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Some Connections between Isoperimetric and Sobolev-type Inequalities

Serguei G. Bobkov Syktyvkar State University, Russia
Christian Houdré Georgia Institute of Technology, Atlanta
Available Formats:
Electronic ISBN: 978-1-4704-0201-3
Product Code: MEMO/129/616.E
List Price: $47.00 MAA Member Price:$42.30
AMS Member Price: $28.20 Click above image for expanded view Some Connections between Isoperimetric and Sobolev-type Inequalities Serguei G. Bobkov Syktyvkar State University, Russia Christian Houdré Georgia Institute of Technology, Atlanta Available Formats:  Electronic ISBN: 978-1-4704-0201-3 Product Code: MEMO/129/616.E  List Price:$47.00 MAA Member Price: $42.30 AMS Member Price:$28.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1291997; 111 pp
MSC: Primary 46; 49; 60; Secondary 32;

For Borel probability measures on metric spaces, the authors study the interplay between isoperimetric and Sobolev-type 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 Sobolev-type inequalities on the real line
• 15. Extensions of Sobolev-type inequalities to product measures on $\mathbf {R}^n$
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1291997; 111 pp
MSC: Primary 46; 49; 60; Secondary 32;

For Borel probability measures on metric spaces, the authors study the interplay between isoperimetric and Sobolev-type 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 Sobolev-type inequalities on the real line
• 15. Extensions of Sobolev-type inequalities to product measures on $\mathbf {R}^n$
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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