A Generalized Central Limit Conjecture for Convex Bodies

TL;DR

This paper introduces a generalized central limit theorem for marginals of isotropic log-concave distributions, establishing a deep connection with the KLS conjecture and its implications for convex geometry.

Contribution

It proposes a generalized CLT for marginals along random directions from any isotropic log-concave distribution, linking it to the KLS conjecture.

Findings

Generalized CLT is equivalent to the KLS conjecture up to small factors

Any improvement in KLS bounds implies a corresponding improvement in the generalized CLT

The results suggest new insights into open problems in convex geometry

Abstract

The central limit theorem for convex bodies says that with high probability the marginal of an isotropic log-concave distribution along a random direction is close to a Gaussian, with the quantitative difference determined asymptotically by the Cheeger/Poincare/KLS constant. Here we propose a generalized CLT for marginals along random directions drawn from any isotropic log-concave distribution; namely, for $x,y$ drawn independently from isotropic log-concave densities $p,q$, the random variable $\langle x,y\rangle$ is close to Gaussian. Our main result is that this generalized CLT is quantitatively equivalent (up to a small factor) to the KLS conjecture. Any polynomial improvement in the current KLS bound of $n^{1/4}$ in $\mathbb{R}^n$ implies the generalized CLT, and vice versa. This tight connection suggests that the generalized CLT might provide insight into basic open questions in asymptotic convex geometry.