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

• 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