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# Some Connections Between Isoperimetric and Sobolev-Type Inequalities

### Memoirs of the American Mathematical Society

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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 ispperimetric 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 sheres, and the canonical Gaussian measure on the Euclidean space.

 Introduction Differential and integral forms of isoperimetric inequalities Proof of Theorem 1.1 A relation between the distribution of a function and its derivative A variational problem The discrete version of Theorem 5.1 Proof of propositions 1.3 and 1.5 A special case of Theorem 1.2 The uniform distribution on the sphere Existence of optimal Orlicz spaces Proof of Theorem 1.9 (the case of the sphere) Proof of Theorem 1.9 (the Gaussian case) The isoperimetric problem on the real line Isoperimetry and Sobolev-type inequalities on the real line Extensions of Sobolev-type inequalities to product measures on $\mathbf{R}^{n}$ References Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780821806425
ISBN-10: 0821806424
Series: Memoirs of the American Mathematical Society : Book 129
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 111
Published: January 1997
Country of Publication: US
Dimensions (cm): 25.4 x 17.8  x 0.64
Weight (kg): 0.25