The Logarithmic Estimate And Equivalent Conditions
Proposition 2.2: Equivalent Conditions
Important result: The following are equivalent:
1. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
2. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
3. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
4. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
5. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
6. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
7. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
8. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
9. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
10. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
11. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
12. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
13. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
14. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
15. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
16. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
17. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
18. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
19. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
20. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
21. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
22. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
23. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
24. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
25. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
26. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
27. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
28. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
29. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
30. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
31. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
32. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
33. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
34. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
35. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
36. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
37. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
38. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
39. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
40. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
41. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
42. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
43. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
44. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
45. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
46. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
47. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
48. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
49. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
50. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
51. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
52. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
53. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
54. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
55. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
56. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
57. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
58. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
59. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
60. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
61. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
62. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
63. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
64. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
65. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
66. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
67. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
68. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
69. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
70. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
71. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
72. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
73. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
74. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
75. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
76. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
77. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
78. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
79. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
80. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
81. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
82. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
83. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
84. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
85. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
86. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
87. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
88. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
89. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
90. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
91. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
92. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
93. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
94. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
95. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
96. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
97. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
98. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
99. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
100. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
101. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
102. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
103. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
104. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
105. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
106. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
107. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
108. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
109. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
110. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
111. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
112. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
113. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
114. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
Important result: The following are equivalent:
115. lim𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 = −1
116. lim𝜖→0liminf𝐴→∞𝐼(𝐴)−1log|𝐴 ∩ 𝛬(1+𝜖)𝐼(𝐴)| = 0
Remark:
• For any set 𝐴, ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ 𝐿(𝐼(𝐴))
• limsup𝐴→∞𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 ≤ −1
• Condition (i) is about the limit inferior as 𝐼(𝐴) tends to infinity
• Condition (ii) is about the limit inferior as 𝜖 tends to 0
Proof of Proposition 2.2
(ii) implies (i)
• Assume (ii) holds
• For any positive 𝜖, derive lower bound for 𝐼(𝐴)−1log ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Show that (i) holds given the remark
(i) implies (ii) (by contradiction)
• Assume (ii) does not hold
• Show existence of positive 𝛽 and sequence 𝜖𝑘→ 0
• Derive upper bound for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Conclude that (i) does not hold
The Basic Bounds
Goal and Approach
Objective: Obtain lower and upper bounds for ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥
• Decompose 𝐴 into small pieces
• Use geometry of the graph of function 𝐼
Assumptions on Function 𝐼
• Strictly convex on ℝ𝑑
• Nonnegative
• Twice differentiable
• lim𝑥→∞𝐼(𝑥)= +∞
• 𝐼(0)= 0 (after translation and scaling)
Important Definitions
• 𝐷𝐼: Differential (gradient) of 𝐼
• 𝐷2𝐼: Hessian of 𝐼 (symmetric definite positive matrix)
• 𝛬𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)≤ 𝑐}: Level set of 𝐼
• 𝛤
𝑐= {𝑥 ∈ ℝ𝑑: 𝐼(𝑥)= 𝑐}: Level line (hypersurface) of 𝐼
Normal Flow
Key concept: Normal flow 𝜓(𝑥, 𝑡)
• Integral curve of the vector field 𝐷𝐼
• 𝜓(𝑥, 0)= 𝑥
• 𝐼(𝜓(𝑥, 𝑡)) = 𝐼(𝑥)+ 𝑡
Definition: The flow 𝜓𝑡 is called the normal flow at time 𝑡.
