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The slicing conjecture via small ball estimates

Pierre Bizeul

TL;DR

The paper targets Bourgain’s slicing conjecture through strong small-ball estimates for isotropic log-concave measures. It proves a dimension-dependent bound showing that, for small ε, the probability that a random vector concentrates within ε n of a point decays like ε to the power c n, by combining stochastic localization with Guan’s covariance bound and a reduction to bounded support. These small-ball bounds are then linked to slicing via Milman’s M-ellipsoid theory, providing an alternative route to a dimension-free slicing constant (complementing the Klartag–Lehec result). The work emphasizes a sequence of reductions and control of measure shrinkage along the localization process to obtain the main inequality and the ensuing slicing bound, highlighting the role of the covariance dynamics.

Abstract

Bourgain's slicing conjecture was recently resolved by Joseph Lehec and Bo'az Klartag. We present an alternative proof by establishing small ball probability estimates for isotropic log-concave measures. Our approach relies on the stochastic localization process and Guan's bound, techniques also used by Klartag and Lehec. The link between small ball probabilities and the slicing conjecture was first observed by Dafnis and Paouris and is established through Milman's theory of M-ellipsoids.

The slicing conjecture via small ball estimates

TL;DR

The paper targets Bourgain’s slicing conjecture through strong small-ball estimates for isotropic log-concave measures. It proves a dimension-dependent bound showing that, for small ε, the probability that a random vector concentrates within ε n of a point decays like ε to the power c n, by combining stochastic localization with Guan’s covariance bound and a reduction to bounded support. These small-ball bounds are then linked to slicing via Milman’s M-ellipsoid theory, providing an alternative route to a dimension-free slicing constant (complementing the Klartag–Lehec result). The work emphasizes a sequence of reductions and control of measure shrinkage along the localization process to obtain the main inequality and the ensuing slicing bound, highlighting the role of the covariance dynamics.

Abstract

Bourgain's slicing conjecture was recently resolved by Joseph Lehec and Bo'az Klartag. We present an alternative proof by establishing small ball probability estimates for isotropic log-concave measures. Our approach relies on the stochastic localization process and Guan's bound, techniques also used by Klartag and Lehec. The link between small ball probabilities and the slicing conjecture was first observed by Dafnis and Paouris and is established through Milman's theory of M-ellipsoids.
Paper Structure (8 sections, 12 theorems, 60 equations)

This paper contains 8 sections, 12 theorems, 60 equations.

Table of Contents

  1. Introduction and result.
  2. Preliminaries
  3. Reduction to small diameter and to $y=0$
  4. The stochastic localization process
  5. Bounding the shrinkage of sets
  6. Proof of the main results
  7. Proof of Theorem \ref{['thm:main']}
  8. Proof of Theorem \ref{['thm_slicing']}

Key Result

Theorem A

Let $X$ be a $b$-subgaussian log-concave vector in $\mathbb R^n$ with covariance matrix $A$. There exist a universal constant $c>0$ such that for any $\varepsilon<c$ and any $y\in\mathbb R^n$,

Theorems & Definitions (17)

  • Theorem A: Paouris
  • Theorem B: Lee-Vampala
  • Theorem 1
  • Lemma 2
  • Theorem C: Klartag-Lehec
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • ...and 7 more