Memoirs of the American Mathematical Society 1997; 111 pp; softcover Volume: 129 ISBN10: 0821806424 ISBN13: 9780821806425 List Price: US$44 Individual Members: US$26.40 Institutional Members: US$35.20 Order Code: MEMO/129/616
 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. Readership Graduate students and research mathematicians interested in probability theory and functional analysis. Table of Contents  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 Sobolevtype inequalities on the real line
 Extensions of Sobolevtype inequalities to product measures on \(\mathbf{R}^{n}\)
 References
