Some Connections between Isoperimetric and Sobolev-type Inequalities

Serguei Germanovich Bobkov · Christian Houdré

1997년 1월 · American Mathematical Society: Memoirs of the American Mathematical Society 616권 · American Mathematical Soc.

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For Borel probability measures on metric spaces, this text studies 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. The book devotes much attention to: the probability distributions on the real line; the normalized Lebesgue measure on the Euclidean sheres; and the canonical Gaussian measure on the Euclidean space.