The slicing conjecture via small ball estimates

TL;DR

This paper offers an alternative proof of Bourgain's slicing conjecture by deriving small ball probability estimates for isotropic log-concave measures using stochastic localization and Guan's bound, connecting small ball probabilities to the conjecture.

Contribution

It introduces a new proof approach for the slicing conjecture leveraging small ball estimates and stochastic localization, differing from prior methods.

Findings

Established small ball probability estimates for isotropic log-concave measures.

Linked small ball probabilities to the slicing conjecture via Milman's M-ellipsoids.

Provided an alternative proof of Bourgain's slicing conjecture.

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.