Asymptotics for Sets Translated Towards Infinity
Introduction
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
1. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
2. Define a candidate for 𝑐𝐴+𝑡,𝑀.
3. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
4. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
5. Define a candidate for 𝑐𝐴+𝑡,𝑀.
6. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
7. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
8. Define a candidate for 𝑐𝐴+𝑡,𝑀.
9. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
10. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
11. Define a candidate for 𝑐𝐴+𝑡,𝑀.
12. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
13. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
14. Define a candidate for 𝑐𝐴+𝑡,𝑀.
15. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
16. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
17. Define a candidate for 𝑐𝐴+𝑡,𝑀.
18. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
19. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
20. Define a candidate for 𝑐𝐴+𝑡,𝑀.
21. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
22. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
23. Define a candidate for 𝑐𝐴+𝑡,𝑀.
24. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
25. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
26. Define a candidate for 𝑐𝐴+𝑡,𝑀.
27. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
28. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
29. Define a candidate for 𝑐𝐴+𝑡,𝑀.
30. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
31. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
32. Define a candidate for 𝑐𝐴+𝑡,𝑀.
33. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
34. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
35. Define a candidate for 𝑐𝐴+𝑡,𝑀.
36. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
37. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
38. Define a candidate for 𝑐𝐴+𝑡,𝑀.
39. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
40. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
41. Define a candidate for 𝑐𝐴+𝑡,𝑀.
42. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
43. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
44. Define a candidate for 𝑐𝐴+𝑡,𝑀.
45. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
46. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
47. Define a candidate for 𝑐𝐴+𝑡,𝑀.
48. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
49. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
50. Define a candidate for 𝑐𝐴+𝑡,𝑀.
51. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
52. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
53. Define a candidate for 𝑐𝐴+𝑡,𝑀.
54. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
55. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
56. Define a candidate for 𝑐𝐴+𝑡,𝑀.
57. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
58. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
59. Define a candidate for 𝑐𝐴+𝑡,𝑀.
60. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
61. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
62. Define a candidate for 𝑐𝐴+𝑡,𝑀.
63. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
64. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
65. Define a candidate for 𝑐𝐴+𝑡,𝑀.
66. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
67. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
68. Define a candidate for 𝑐𝐴+𝑡,𝑀.
69. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
70. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
71. Define a candidate for 𝑐𝐴+𝑡,𝑀.
72. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
73. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
74. Define a candidate for 𝑐𝐴+𝑡,𝑀.
75. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
76. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
77. Define a candidate for 𝑐𝐴+𝑡,𝑀.
78. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
79. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
80. Define a candidate for 𝑐𝐴+𝑡,𝑀.
81. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
82. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
83. Define a candidate for 𝑐𝐴+𝑡,𝑀.
84. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
85. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
86. Define a candidate for 𝑐𝐴+𝑡,𝑀.
87. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
88. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
89. Define a candidate for 𝑐𝐴+𝑡,𝑀.
90. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
91. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
92. Define a candidate for 𝑐𝐴+𝑡,𝑀.
93. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
94. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
95. Define a candidate for 𝑐𝐴+𝑡,𝑀.
96. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
97. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
98. Define a candidate for 𝑐𝐴+𝑡,𝑀.
99. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
100. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
101. Define a candidate for 𝑐𝐴+𝑡,𝑀.
102. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
103. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
104. Define a candidate for 𝑐𝐴+𝑡,𝑀.
105. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
106. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
107. Define a candidate for 𝑐𝐴+𝑡,𝑀.
108. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
109. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
110. Define a candidate for 𝑐𝐴+𝑡,𝑀.
111. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
112. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
113. Define a candidate for 𝑐𝐴+𝑡,𝑀.
114. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
115. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
116. Define a candidate for 𝑐𝐴+𝑡,𝑀.
117. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
118. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
119. Define a candidate for 𝑐𝐴+𝑡,𝑀.
120. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
121. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
122. Define a candidate for 𝑐𝐴+𝑡,𝑀.
123. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
124. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
125. Define a candidate for 𝑐𝐴+𝑡,𝑀.
126. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
127. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
128. Define a candidate for 𝑐𝐴+𝑡,𝑀.
129. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
130. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
131. Define a candidate for 𝑐𝐴+𝑡,𝑀.
132. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
133. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
134. Define a candidate for 𝑐𝐴+𝑡,𝑀.
135. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
136. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
137. Define a candidate for 𝑐𝐴+𝑡,𝑀.
138. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
139. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
140. Define a candidate for 𝑐𝐴+𝑡,𝑀.
141. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
142. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
143. Define a candidate for 𝑐𝐴+𝑡,𝑀.
144. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
145. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
146. Define a candidate for 𝑐𝐴+𝑡,𝑀.
147. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
148. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
149. Define a candidate for 𝑐𝐴+𝑡,𝑀.
150. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
151. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
152. Define a candidate for 𝑐𝐴+𝑡,𝑀.
153. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
154. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
155. Define a candidate for 𝑐𝐴+𝑡,𝑀.
156. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
157. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
158. Define a candidate for 𝑐𝐴+𝑡,𝑀.
159. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
160. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
161. Define a candidate for 𝑐𝐴+𝑡,𝑀.
162. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
163. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
164. Define a candidate for 𝑐𝐴+𝑡,𝑀.
165. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
166. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
167. Define a candidate for 𝑐𝐴+𝑡,𝑀.
168. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
169. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
170. Define a candidate for 𝑐𝐴+𝑡,𝑀.
171. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
172. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
173. Define a candidate for 𝑐𝐴+𝑡,𝑀.
174. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
175. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
176. Define a candidate for 𝑐𝐴+𝑡,𝑀.
177. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
178. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
179. Define a candidate for 𝑐𝐴+𝑡,𝑀.
180. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
181. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
182. Define a candidate for 𝑐𝐴+𝑡,𝑀.
183. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
184. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
185. Define a candidate for 𝑐𝐴+𝑡,𝑀.
186. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
187. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
188. Define a candidate for 𝑐𝐴+𝑡,𝑀.
189. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
190. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
191. Define a candidate for 𝑐𝐴+𝑡,𝑀.
192. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
193. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
194. Define a candidate for 𝑐𝐴+𝑡,𝑀.
195. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
196. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
197. Define a candidate for 𝑐𝐴+𝑡,𝑀.
198. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
199. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
200. Define a candidate for 𝑐𝐴+𝑡,𝑀.
201. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
202. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
203. Define a candidate for 𝑐𝐴+𝑡,𝑀.
204. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
205. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
206. Define a candidate for 𝑐𝐴+𝑡,𝑀.
207. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
208. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
209. Define a candidate for 𝑐𝐴+𝑡,𝑀.
210. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
211. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
212. Define a candidate for 𝑐𝐴+𝑡,𝑀.
213. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
214. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
215. Define a candidate for 𝑐𝐴+𝑡,𝑀.
216. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
217. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
218. Define a candidate for 𝑐𝐴+𝑡,𝑀.
219. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
220. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
221. Define a candidate for 𝑐𝐴+𝑡,𝑀.
222. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
223. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
224. Define a candidate for 𝑐𝐴+𝑡,𝑀.
225. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
226. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
227. Define a candidate for 𝑐𝐴+𝑡,𝑀.
228. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
229. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
230. Define a candidate for 𝑐𝐴+𝑡,𝑀.
231. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
232. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
233. Define a candidate for 𝑐𝐴+𝑡,𝑀.
234. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
235. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
236. Define a candidate for 𝑐𝐴+𝑡,𝑀.
237. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
238. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
239. Define a candidate for 𝑐𝐴+𝑡,𝑀.
240. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
241. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
242. Define a candidate for 𝑐𝐴+𝑡,𝑀.
243. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
244. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
245. Define a candidate for 𝑐𝐴+𝑡,𝑀.
246. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
247. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
248. Define a candidate for 𝑐𝐴+𝑡,𝑀.
249. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
These lecture notes discuss the asymptotic behavior of integrals of the form ∫𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥
as |𝑡| tends to infinity.
Assumptions
• 𝐴⊂ℝ𝑑 is a closed bounded convex neighborhood of the origin, with smooth
boundary and positive curvature.
• The function 𝐼 satisfies:
lim
𝑝→∞log𝐼(𝑝)
|𝐷𝐼(𝑝)|=0 and lim
𝑝→∞∥𝐷2𝐼(𝑝)∥
|𝐷𝐼(𝑝)| =0
• Additional condition:
lim
𝑡→∞log𝐼(𝐴+𝑡)sup{∥𝐷2𝐼(𝑞)∥
|𝐷𝐼(𝑞)| :𝐼(𝑞)>𝐼(𝐴+𝑡)}=0
Main Theorem
Theorem 6.1: If 𝐼 is convex and the above assumptions hold, then
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑝𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
where 𝐾𝑡 is the Gauss-Kronecker curvature of ∂𝐴 at 𝑝𝑡−𝑡.
Proof Outline
250. Apply Theorem 5.1 with 𝐷𝐴𝑡={𝑝𝑡}.
251. Define a candidate for 𝑐𝐴+𝑡,𝑀.
252. Check all assumptions of Theorem 5.1:
– Trivial conditions: (5.1), (5.7)
– Geometric conditions: (5.2), (5.3)
– Shrinking of 𝐴𝑡,𝑀 around 𝑝𝑡: (5.5)
– Curvature and normal flow approximations: (5.8), (5.9), (5.10)
– Technical conditions: (5.11), (5.12), (5.13)
Corollary
Corollary 6.2: Under the same assumptions,
∫ 𝑒−𝐼(𝑥)
𝐴+𝑡 𝑑𝑥 ∼ (2𝜋)(𝑑−1)/2
|𝐷𝐼(𝑡)|(𝑑+1)/2𝐾𝑡𝑒−𝐼(𝐴+𝑡) as 𝑡→∞
Key Techniques
• Normal flow approximation
• Linear approximation of the exponential map
• Uniform estimates on derivatives of 𝐼
• Geometric properties of convex sets and their level surfaces
