Entropy bounds and isoperimetry

S. G. Bobkov, B. Zegarlinski

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37 Scopus citations

Abstract

We consider coercive and isoperimetric inequalities for probability measures with super-Gaussian tails. First part of the work is devoted to analysis of modified forms of logarithmic Sobolev inequalities, which we call LSq, on abstract (metric) spaces with respect to Lq-norm of the modulus of gradient. We give an explicit characterization of probability distributions satisfying these inequalities on the real line. It is also shown that LSg are satisfied for a large class of non-trivial infinite dimensional measures (such as Gibbs measures). In the second part, Sobolev-type inequalities are studied in the class of uniform distributions on convex bodies in finite dimensional Euclidean spaces. In particular, we refine a result of Kannan, Lovász and Simonovits on the Cheeger isoperimetric constants.

Original languageEnglish (US)
JournalMemoirs of the American Mathematical Society
Volume176
Issue number829
DOIs
StatePublished - Jul 2005

Keywords

  • Coercive and isoperimetric inequalities
  • Concentration of measure
  • Gibbs measures
  • Infinite dimensional spaces
  • Markov semigroups

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