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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
Front Cover for Some Connections between Isoperimetric and Sobolev-type Inequalities
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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
Front Cover for Some Connections between Isoperimetric and Sobolev-type Inequalities
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  • Front Cover for Some Connections between Isoperimetric and Sobolev-type Inequalities
  • Back Cover for Some Connections between Isoperimetric and Sobolev-type Inequalities
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.

    Readership

    Graduate 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 Sobolev-type inequalities on the real line
    • 15. Extensions of Sobolev-type inequalities to product measures on $\mathbf {R}^n$
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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.

Readership

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 $\mathbf {R}^n$
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