A note on Bourgain's slicing problem

TL;DR

The paper proves an iterated logarithm bound for Bourgain's slicing constant, showing $L_n \le C\log(\log n)$ for $n\ge 3$ by tying $L_n$ to the thin cell constant $\sigma_n$ and the isoperimetric constant $\psi_n$ via the Eldan–Klartag framework. It develops a nonhomogeneous Γ-calculus/stochastic localization approach, introducing auxiliary test functions and stochastic processes to control the covariance evolution $A_t$ and bound $\mathbb{E}\mathrm{Tr}(A_t^2)$. A key step is establishing $\sigma_\mu \le C|\log\psi_\mu|+C$ for isotropic log-concave $\mu$, and then leveraging Klartag's bound $\psi_n \le C\sqrt{\log n}$ to obtain the main result. This work clarifies the intricate connections between slicing, thin-cell, and KLS-type isoperimetric conjectures, and situates the iterated logarithm bound within the broader landscape of high-dimensional convex geometry.

Abstract

This note is to study Bourgain's slicing problem following the routes investigated in the last decade. We show that the slicing constant $L_n$ is bounded by $C\log(\log n) $, $n\geq 3$, for some universal constant $C$.