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A Slightly Improved Bound for the KLS Constant

Arun Jambulapati, Yin Tat Lee, Santosh S. Vempala

TL;DR

This work advances the study of isotropic log-concave densities by sharpening the polylogarithmic bounds on the thin-shell and KLS constants. Building on the Klartag–Lehec framework, it refines stochastic localization via the LV process and develops sharper tensor bounds for $T_{\mu}(I,I,I)$, connecting mean dynamics to concentration width. The authors prove an unconditional bound $T_{p}(I,I,I)$ with $\gamma\le 2\sqrt{2}$, establish a conditional regime yielding further improvements when the covariance is suitably structured, and show a sub-logarithmic bound under a no-gap regime $\psi_{n}\approx\sigma_{n}$, culminating in $\sigma_{n}\lesssim\log^{2.2226}n$ and $\psi_{n}\lesssim\log^{3.2226}n$. These results push toward tighter, dimension-free understanding of the KLS constant and highlight new stochastic-decomposition tools that may have broader applicability in convex-geometry and high-dimensional probability.

Abstract

We refine the recent breakthrough technique of Klartag and Lehec to obtain an improved polylogarithmic bound for the KLS constant.

A Slightly Improved Bound for the KLS Constant

TL;DR

This work advances the study of isotropic log-concave densities by sharpening the polylogarithmic bounds on the thin-shell and KLS constants. Building on the Klartag–Lehec framework, it refines stochastic localization via the LV process and develops sharper tensor bounds for , connecting mean dynamics to concentration width. The authors prove an unconditional bound with , establish a conditional regime yielding further improvements when the covariance is suitably structured, and show a sub-logarithmic bound under a no-gap regime , culminating in and . These results push toward tighter, dimension-free understanding of the KLS constant and highlight new stochastic-decomposition tools that may have broader applicability in convex-geometry and high-dimensional probability.

Abstract

We refine the recent breakthrough technique of Klartag and Lehec to obtain an improved polylogarithmic bound for the KLS constant.
Paper Structure (15 sections, 24 theorems, 119 equations)

This paper contains 15 sections, 24 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Discussion of the Klartag-Lehec bound
  4. Our contributions
  5. Unconditional bound on $T_{p}(I,I,I)$.
  6. Conditional bound on $T_{p}(I,I,I)$.
  7. Sub-logarithmic bound assuming $\psi_{n}\approx\sigma_{n}$.
  8. Preliminaries
  9. Bounding $T_{\mu}(I,I,I)$
  10. Unconditional Bound
  11. Conditional Bound
  12. Bounding $\mathrm{Tr} A_{t}^{q}$
  13. Improved bounds for thin shell
  14. Discussion on possible value of $\gamma$
  15. Deferred calculations

Key Result

Theorem 1

$\sigma_{n}\lesssim\log^{4}n$, $\psi_{n}\lesssim\log^{5}n.$Throughout this paper, we use $a\lesssim b$ to denote $a=O(b)$ and $a\approx b$ to denote $a=\Theta(b)$.

Theorems & Definitions (40)

  • Theorem 1: klartag2022bourgain
  • Theorem 2
  • Definition 3: Constants
  • Definition 4: Stochastic localization
  • Lemma 5: Eldan2013
  • Definition 6
  • Lemma 7: chen2021almost
  • Lemma 8: klartag2022bourgain
  • Lemma 9: cordero2004b
  • Lemma 10: Properties of the Eldan process
  • ...and 30 more