Electronic ISBN:  9781470402013 
Product Code:  MEMO/129/616.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 129; 1997; 111 ppMSC: Primary 46; 49; 60; Secondary 32;
For Borel probability measures on metric spaces, the authors study the interplay between isoperimetric and Sobolevtype inequalities. In particular the question of finding optimal constants via isoperimetric quantities is explored. Also given are necessary and sufficient conditions for the equivalence between the extremality of some sets in the isoperimetric problem and the validity of some analytic inequalities. Much attention is devoted to probability distributions on the real line, the normalized Lebesgue measure on the Euclidean spheres, and the canonical Gaussian measure on the Euclidean space.
ReadershipGraduate students and research mathematicians interested in probability theory and functional analysis.

Table of Contents

Chapters

1. Introduction

2. Differential and integral forms of isoperimetric inequalities

3. Proof of Theorem 1.1

4. A relation between the distribution of a function and its derivative

5. A variational problem

6. The discrete version of Theorem 5.1

7. Proof of Propositions 1.3 and 1.5

8. A special case of Theorem 1.2

9. The uniform distribution on the sphere

10. Existence of optimal Orlicz spaces

11. Proof of Theorem 1.9 (the case of the sphere)

12. Proof of Theorem 1.9 (the Gaussian case)

13. The isoperimetric problem on the real line

14. Isoperimetric and Sobolevtype inequalities on the real line

15. Extensions of Sobolevtype inequalities to product measures on $\mathbf {R}^n$


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For Borel probability measures on metric spaces, the authors study the interplay between isoperimetric and Sobolevtype inequalities. In particular the question of finding optimal constants via isoperimetric quantities is explored. Also given are necessary and sufficient conditions for the equivalence between the extremality of some sets in the isoperimetric problem and the validity of some analytic inequalities. Much attention is devoted to probability distributions on the real line, the normalized Lebesgue measure on the Euclidean spheres, and the canonical Gaussian measure on the Euclidean space.
Graduate students and research mathematicians interested in probability theory and functional analysis.

Chapters

1. Introduction

2. Differential and integral forms of isoperimetric inequalities

3. Proof of Theorem 1.1

4. A relation between the distribution of a function and its derivative

5. A variational problem

6. The discrete version of Theorem 5.1

7. Proof of Propositions 1.3 and 1.5

8. A special case of Theorem 1.2

9. The uniform distribution on the sphere

10. Existence of optimal Orlicz spaces

11. Proof of Theorem 1.9 (the case of the sphere)

12. Proof of Theorem 1.9 (the Gaussian case)

13. The isoperimetric problem on the real line

14. Isoperimetric and Sobolevtype inequalities on the real line

15. Extensions of Sobolevtype inequalities to product measures on $\mathbf {R}^n$