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A note on the spectral gap for log-concave probability measures on convex bodies

Michel Bonnefont, Aldéric Joulin

Abstract

In this paper, we provide explicit lower bounds with respect to some quantities of interest (parameters of the underlying distribution, dimension, geometrical characteristics of the domain, position of the origin, etc.) on the spectral gap of log-concave probability measures on convex bodies. Our results are illustrated by some classical and less classical examples.

A note on the spectral gap for log-concave probability measures on convex bodies

Abstract

In this paper, we provide explicit lower bounds with respect to some quantities of interest (parameters of the underlying distribution, dimension, geometrical characteristics of the domain, position of the origin, etc.) on the spectral gap of log-concave probability measures on convex bodies. Our results are illustrated by some classical and less classical examples.
Paper Structure (12 sections, 9 theorems, 123 equations)

This paper contains 12 sections, 9 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminary results and proof of Theorem \ref{['theo:main']}
  3. Basic material and notation
  4. Brascamp-Lieb type inequalities and general spectral gap estimates
  5. Proof of Theorem \ref{['theo:main']}
  6. A class of convex bodies
  7. General log-concave probability measures
  8. The uniform distribution
  9. The uniformly convex case
  10. The Subbotin distribution
  11. The case of generalized Orlicz balls
  12. Appendix

Key Result

Theorem 1.1

On a (connected) compact set $\Omega \subset {\mathbb{R}}^d$ ($d\geq 2$) with smooth boundary $\partial \Omega$ and outer unit-normal $\eta$, we consider a probability measure $\mu$ whose Lebesgue density is proportional to $e^{-V}$, where $V :\Omega \to {\mathbb{R}}$ is some sufficiently smooth pot Then the generalized Brascamp-Lieb inequality holds: for all $g \in {\mathcal{C}\ \!\!} ^\infty (\O

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['theo:main']}
  • Corollary 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 8 more