Spectral gap for some invariant log-concave probability measures

TL;DR

This paper proves the Kannan-Lovász-Simonovits conjecture for certain invariant log-concave measures, including those associated with convex bodies of revolution, by establishing spectral gap properties.

Contribution

It extends the conjecture's validity to log-concave measures of specific forms, notably those invariant under certain symmetries, and confirms it for convex bodies of revolution.

Findings

The conjecture holds for measures of the form ρ(|x|_B)dx.

It is valid for measures with densities ρ(t,|x|_B) dx.

Confirmed the conjecture for convex bodies of revolution.

Abstract

We show that the conjecture of Kannan, Lov\'{a}sz, and Simonovits on isoperimetric properties of convex bodies and log-concave measures, is true for log-concave measures of the form $\rho(|x|_B)dx$ on $\mathbb{R}^n$ and $\rho(t,|x|_B) dx$ on $\mathbb{R}^{1+n}$, where $|x|_B$ is the norm associated to any convex body $B$ already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.