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On the log-Sobolev constant of log-concave measures

Pierre Bizeul

TL;DR

The paper advances understanding of how log-concave measures with subgaussian tails satisfy log-Sobolev inequalities by establishing a sharper, dimension-dependent bound $G_n \le C\,n^{1/4}$ for the log-Sobolev constant among $1$-subgaussian log-concave measures. The core strategy blends localization (reducing to one dimension) with stochastic localization to control concentration and derive LSI bounds; it also develops a tilt-stability framework that links perturbations of tilts to subgaussian behavior and LSI constants. Key contributions include a 1D sharp bound $\rho_{LS}^2(X) \lesssim \|X^2-\mathbb{E}X^2\|_{\psi_1}$, a geometric localization lemma connecting $k_\mu$ to $\|\lvert X\rvert^2-a\|_{\psi_1}$, and a perturbation/strong tilt-stability theory showing $1$-tilt-stable log-concave measures are strongly tilt-stable with quantitative dependence on dimension and a coordinate-psi2 norm. Collectively, these results provide new pathways toward the KLS conjecture by bounding the log-Sobolev constant through subgaussian coordinates and tilt-perturbation analyses.

Abstract

It is well known that a log-Sobolev inequality implies sub-gaussian decay of the tails. In the spirit of the KLS conjecture, we investigate whether this implication can be reversed under a log-concavity assumption. In the general setting, we improve on a result of Bobkov, establishing the best dimension dependent bound on the log-Sobolev constant of subgaussian log-concave measures, and we investigate some special cases.

On the log-Sobolev constant of log-concave measures

TL;DR

The paper advances understanding of how log-concave measures with subgaussian tails satisfy log-Sobolev inequalities by establishing a sharper, dimension-dependent bound for the log-Sobolev constant among -subgaussian log-concave measures. The core strategy blends localization (reducing to one dimension) with stochastic localization to control concentration and derive LSI bounds; it also develops a tilt-stability framework that links perturbations of tilts to subgaussian behavior and LSI constants. Key contributions include a 1D sharp bound , a geometric localization lemma connecting to , and a perturbation/strong tilt-stability theory showing -tilt-stable log-concave measures are strongly tilt-stable with quantitative dependence on dimension and a coordinate-psi2 norm. Collectively, these results provide new pathways toward the KLS conjecture by bounding the log-Sobolev constant through subgaussian coordinates and tilt-perturbation analyses.

Abstract

It is well known that a log-Sobolev inequality implies sub-gaussian decay of the tails. In the spirit of the KLS conjecture, we investigate whether this implication can be reversed under a log-concavity assumption. In the general setting, we improve on a result of Bobkov, establishing the best dimension dependent bound on the log-Sobolev constant of subgaussian log-concave measures, and we investigate some special cases.
Paper Structure (15 sections, 33 theorems, 154 equations)

This paper contains 15 sections, 33 theorems, 154 equations.

Table of Contents

  1. Introduction and results
  2. Background and Proof of Theorem \ref{['thm:MAINLOGSOB']}
  3. Sub-gaussian random variables
  4. Sub-exponential random variables
  5. Log-concave vectors
  6. A short proof of Theorem \ref{['thm:MAINLOGSOB']}
  7. Proof of Theorem \ref{['thm:LOCALISATIONLOGSOB']}
  8. The one dimensional case
  9. A localization argument
  10. An approach via stochastic localization
  11. Some subclasses of subgaussian log-concave probabilities
  12. Rotationally invariant measures
  13. Tilt-stable measures
  14. A perturbation result
  15. Proof of Lemma \ref{['lem:intro_tilt_implies_strong_tilt']} and Theorem \ref{['thm:tilt_logsob']}

Key Result

Theorem 4

There exists a constant $C>0$ such that for all $n\geq1$

Theorems & Definitions (57)

  • Conjecture 1: KLS
  • Conjecture 2
  • Definition 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Lemma 8
  • Conjecture 9
  • Theorem 10
  • ...and 47 more