Logarithmic Estimate for Exponential Integrals
Introduction
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
1. Use Fubini’s theorem and change of variable
2. Utilize convexity of 𝐼
3. Bound |𝛬𝑡| using geometric properties
4. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
1. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
2. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
3. Use Fubini’s theorem and change of variable
4. Utilize convexity of 𝐼
5. Bound |𝛬𝑡| using geometric properties
6. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
7. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
8. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
9. Use Fubini’s theorem and change of variable
10. Utilize convexity of 𝐼
11. Bound |𝛬𝑡| using geometric properties
12. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
13. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
14. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
15. Use Fubini’s theorem and change of variable
16. Utilize convexity of 𝐼
17. Bound |𝛬𝑡| using geometric properties
18. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
19. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
20. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
21. Use Fubini’s theorem and change of variable
22. Utilize convexity of 𝐼
23. Bound |𝛬𝑡| using geometric properties
24. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
25. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
26. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
27. Use Fubini’s theorem and change of variable
28. Utilize convexity of 𝐼
29. Bound |𝛬𝑡| using geometric properties
30. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
31. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
32. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
33. Use Fubini’s theorem and change of variable
34. Utilize convexity of 𝐼
35. Bound |𝛬𝑡| using geometric properties
36. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
37. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
38. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
39. Use Fubini’s theorem and change of variable
40. Utilize convexity of 𝐼
41. Bound |𝛬𝑡| using geometric properties
42. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
43. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
44. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
45. Use Fubini’s theorem and change of variable
46. Utilize convexity of 𝐼
47. Bound |𝛬𝑡| using geometric properties
48. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
49. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
50. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
51. Use Fubini’s theorem and change of variable
52. Utilize convexity of 𝐼
53. Bound |𝛬𝑡| using geometric properties
54. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
55. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
56. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
57. Use Fubini’s theorem and change of variable
58. Utilize convexity of 𝐼
59. Bound |𝛬𝑡| using geometric properties
60. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
61. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
62. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
63. Use Fubini’s theorem and change of variable
64. Utilize convexity of 𝐼
65. Bound |𝛬𝑡| using geometric properties
66. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
67. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
68. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
69. Use Fubini’s theorem and change of variable
70. Utilize convexity of 𝐼
71. Bound |𝛬𝑡| using geometric properties
72. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
73. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
74. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
75. Use Fubini’s theorem and change of variable
76. Utilize convexity of 𝐼
77. Bound |𝛬𝑡| using geometric properties
78. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
79. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
80. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
81. Use Fubini’s theorem and change of variable
82. Utilize convexity of 𝐼
83. Bound |𝛬𝑡| using geometric properties
84. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
85. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
86. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
87. Use Fubini’s theorem and change of variable
88. Utilize convexity of 𝐼
89. Bound |𝛬𝑡| using geometric properties
90. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
91. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
92. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
93. Use Fubini’s theorem and change of variable
94. Utilize convexity of 𝐼
95. Bound |𝛬𝑡| using geometric properties
96. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
97. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
98. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
99. Use Fubini’s theorem and change of variable
100. Utilize convexity of 𝐼
101. Bound |𝛬𝑡| using geometric properties
102. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
103. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
104. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
105. Use Fubini’s theorem and change of variable
106. Utilize convexity of 𝐼
107. Bound |𝛬𝑡| using geometric properties
108. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
109. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
110. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
111. Use Fubini’s theorem and change of variable
112. Utilize convexity of 𝐼
113. Bound |𝛬𝑡| using geometric properties
114. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
115. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
116. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
117. Use Fubini’s theorem and change of variable
118. Utilize convexity of 𝐼
119. Bound |𝛬𝑡| using geometric properties
120. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
121. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
122. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
123. Use Fubini’s theorem and change of variable
124. Utilize convexity of 𝐼
125. Bound |𝛬𝑡| using geometric properties
126. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
127. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
128. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
129. Use Fubini’s theorem and change of variable
130. Utilize convexity of 𝐼
131. Bound |𝛬𝑡| using geometric properties
132. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
133. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
134. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
135. Use Fubini’s theorem and change of variable
136. Utilize convexity of 𝐼
137. Bound |𝛬𝑡| using geometric properties
138. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
139. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
140. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
141. Use Fubini’s theorem and change of variable
142. Utilize convexity of 𝐼
143. Bound |𝛬𝑡| using geometric properties
144. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
145. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
146. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
147. Use Fubini’s theorem and change of variable
148. Utilize convexity of 𝐼
149. Bound |𝛬𝑡| using geometric properties
150. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
151. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
152. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
153. Use Fubini’s theorem and change of variable
154. Utilize convexity of 𝐼
155. Bound |𝛬𝑡| using geometric properties
156. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
157. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
158. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
159. Use Fubini’s theorem and change of variable
160. Utilize convexity of 𝐼
161. Bound |𝛬𝑡| using geometric properties
162. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
163. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
164. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
165. Use Fubini’s theorem and change of variable
166. Utilize convexity of 𝐼
167. Bound |𝛬𝑡| using geometric properties
168. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
169. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
170. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
171. Use Fubini’s theorem and change of variable
172. Utilize convexity of 𝐼
173. Bound |𝛬𝑡| using geometric properties
174. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
175. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
176. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
177. Use Fubini’s theorem and change of variable
178. Utilize convexity of 𝐼
179. Bound |𝛬𝑡| using geometric properties
180. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
181. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
182. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
183. Use Fubini’s theorem and change of variable
184. Utilize convexity of 𝐼
185. Bound |𝛬𝑡| using geometric properties
186. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
187. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
188. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
189. Use Fubini’s theorem and change of variable
190. Utilize convexity of 𝐼
191. Bound |𝛬𝑡| using geometric properties
192. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
193. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
194. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
195. Use Fubini’s theorem and change of variable
196. Utilize convexity of 𝐼
197. Bound |𝛬𝑡| using geometric properties
198. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
199. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
200. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
201. Use Fubini’s theorem and change of variable
202. Utilize convexity of 𝐼
203. Bound |𝛬𝑡| using geometric properties
204. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
205. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
206. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
207. Use Fubini’s theorem and change of variable
208. Utilize convexity of 𝐼
209. Bound |𝛬𝑡| using geometric properties
210. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
211. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
212. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
213. Use Fubini’s theorem and change of variable
214. Utilize convexity of 𝐼
215. Bound |𝛬𝑡| using geometric properties
216. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
217. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
218. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
219. Use Fubini’s theorem and change of variable
220. Utilize convexity of 𝐼
221. Bound |𝛬𝑡| using geometric properties
222. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
223. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
224. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
225. Use Fubini’s theorem and change of variable
226. Utilize convexity of 𝐼
227. Bound |𝛬𝑡| using geometric properties
228. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
229. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
230. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
231. Use Fubini’s theorem and change of variable
232. Utilize convexity of 𝐼
233. Bound |𝛬𝑡| using geometric properties
234. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
235. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
236. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
237. Use Fubini’s theorem and change of variable
238. Utilize convexity of 𝐼
239. Bound |𝛬𝑡| using geometric properties
240. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
241. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
242. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
243. Use Fubini’s theorem and change of variable
244. Utilize convexity of 𝐼
245. Bound |𝛬𝑡| using geometric properties
246. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
247. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
248. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
249. Use Fubini’s theorem and change of variable
250. Utilize convexity of 𝐼
251. Bound |𝛬𝑡| using geometric properties
252. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
253. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
254. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
255. Use Fubini’s theorem and change of variable
256. Utilize convexity of 𝐼
257. Bound |𝛬𝑡| using geometric properties
258. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
259. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
260. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
261. Use Fubini’s theorem and change of variable
262. Utilize convexity of 𝐼
263. Bound |𝛬𝑡| using geometric properties
264. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
265. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
266. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
267. Use Fubini’s theorem and change of variable
268. Utilize convexity of 𝐼
269. Bound |𝛬𝑡| using geometric properties
270. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
271. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
272. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
273. Use Fubini’s theorem and change of variable
274. Utilize convexity of 𝐼
275. Bound |𝛬𝑡| using geometric properties
276. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
277. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
278. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
279. Use Fubini’s theorem and change of variable
280. Utilize convexity of 𝐼
281. Bound |𝛬𝑡| using geometric properties
282. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
283. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
284. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
285. Use Fubini’s theorem and change of variable
286. Utilize convexity of 𝐼
287. Bound |𝛬𝑡| using geometric properties
288. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
289. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
290. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
291. Use Fubini’s theorem and change of variable
292. Utilize convexity of 𝐼
293. Bound |𝛬𝑡| using geometric properties
294. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
295. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
296. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
297. Use Fubini’s theorem and change of variable
298. Utilize convexity of 𝐼
299. Bound |𝛬𝑡| using geometric properties
300. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
301. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
302. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
303. Use Fubini’s theorem and change of variable
304. Utilize convexity of 𝐼
305. Bound |𝛬𝑡| using geometric properties
306. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
307. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
308. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
309. Use Fubini’s theorem and change of variable
310. Utilize convexity of 𝐼
311. Bound |𝛬𝑡| using geometric properties
312. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
313. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
314. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
315. Use Fubini’s theorem and change of variable
316. Utilize convexity of 𝐼
317. Bound |𝛬𝑡| using geometric properties
318. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
319. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
320. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
321. Use Fubini’s theorem and change of variable
322. Utilize convexity of 𝐼
323. Bound |𝛬𝑡| using geometric properties
324. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
325. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
326. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
327. Use Fubini’s theorem and change of variable
328. Utilize convexity of 𝐼
329. Bound |𝛬𝑡| using geometric properties
330. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
331. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
332. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
333. Use Fubini’s theorem and change of variable
334. Utilize convexity of 𝐼
335. Bound |𝛬𝑡| using geometric properties
336. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
337. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
338. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
339. Use Fubini’s theorem and change of variable
340. Utilize convexity of 𝐼
341. Bound |𝛬𝑡| using geometric properties
342. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
343. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
344. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
345. Use Fubini’s theorem and change of variable
346. Utilize convexity of 𝐼
347. Bound |𝛬𝑡| using geometric properties
348. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
349. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
350. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
351. Use Fubini’s theorem and change of variable
352. Utilize convexity of 𝐼
353. Bound |𝛬𝑡| using geometric properties
354. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
355. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
356. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
• We consider a nonnegative function 𝐼 defined on ℝ𝑑
• 𝐼 is convex and lim|𝑥|→∞𝐼(𝑥)= +∞
• For any subset 𝐴 of ℝ𝑑, we define 𝐼(𝐴)=inf{𝐼(𝑥):𝑥 ∈ 𝐴}
Main Question
Is ∫𝑒−𝐼(𝑥)
𝐴𝑑𝑥 close to 𝑒−𝐼(𝐴) logarithmically when 𝐴 is far from the origin?
For sets 𝐴 with smooth boundary:
log∫exp
𝐴(−𝜆𝐼(𝑥))𝑑𝑥 ∼ 𝜆𝐼(𝐴) as 𝜆 → ∞
Upper Estimate for Complements of Level Sets
Level Sets of 𝐼
𝛬𝑐= {𝑥:𝐼(𝑥)≤ 𝑐}, 𝑐 > 0
Function 𝐿(𝑐)
𝐿(𝑐)= ∫ 𝑒−𝐼(𝑥)
𝛬𝑐
𝑐𝑑𝑥, 𝑐 > 0
Proposition 2.1: There exists a constant 𝐶0 such that for any positive 𝑐, 𝐿(𝑐)≤
𝐶0𝑒−𝑐(1+𝑐)𝑑
Proof Outline
357. Use Fubini’s theorem and change of variable
358. Utilize convexity of 𝐼
359. Bound |𝛬𝑡| using geometric properties
360. Apply integral inequalities
Notation for Limits
For functions 𝐹,𝐺 defined on the Borel 𝜎-field of ℝ𝑑:
361. lim𝐴→∞𝐹(𝐴)= 0 if and only if:
∀𝜖 > 0,∃𝑐:𝐼(𝐴)> 𝑐 ⟹ |𝐹(𝐴)|≤ 𝜖
362. lim𝐴→∞𝐹(𝐴)= 0 ⟹ lim𝐴→∞𝐺(𝐴)= 0 if and only if:
∀𝜖,∃𝛿,∃𝑐 > 0:(𝐼(𝐴)> 𝑐 and |𝐹(𝐴)| ≤ 𝛿)⟹|𝐺(𝐴)| ≤ 𝜖
Note: This notation allows expressing approximation properties for sets moving away from
the origin under analytical constraints.
