Remarks on the KLS conjecture and Hardy-type inequalities

TL;DR

This paper extends classical inequalities to general functions on convex bodies, linking geometric properties to spectral constants, and proves a conjecture related to log-concave measures and convex bodies.

Contribution

It generalizes Hardy and Faber-Krahn inequalities to broader function classes on convex bodies and proves a key conjecture for specific convex bodies.

Findings

Reduced study of Neumann Poincaré constant to boundary measures and curvature.

Established bounds for Poincaré constants using geometric properties.

Provided a simple proof of the Kannan-Lovász-Simonovits conjecture for $ ext{l}^n_p$ balls.

Abstract

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body $\Omega \subset \mathbb{R}^n$, not necessarily vanishing on the boundary $\partial \Omega$. This reduces the study of the Neumann Poincar\'e constant on $\Omega$ to that of the cone and Lebesgue measures on $\partial \Omega$; these may be bounded via the curvature of $\partial \Omega$. A second reduction is obtained to the class of harmonic functions on $\Omega$. We also study the relation between the Poincar\'e constant of a log-concave measure $\mu$ and its associated K. Ball body $K_\mu$. In particular, we obtain a simple proof of a conjecture of Kannan--Lov\'asz--Simonovits for unit-balls of $\ell^n_p$, originally due to Sodin and Lata{\l}a--Wojtaszczyk.