Remarks on the KLS conjecture and Hardy-type inequalities

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 Poincaré 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 Poincaré 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--Lovász--Simonovits for unit-balls of ${\ell }_p^n$, originally due to Sodin and Latała--Wojtaszczyk.

18 pages. Numbering of propositions, theorems, etc.. as appeared in final form in GAFA seminar notes

References in corpus (2)

• Thin shell implies spectral gap up to polylog via a stochastic localization scheme
• An isoperimetric inequality on the ${\ell }_p$ balls