Improved log-concavity for rotationally invariant measures of symmetric convex sets

TL;DR

This paper proves the B conjecture and Gardner-Zvavitch conjecture for all rotationally invariant log-concave measures, extending known results beyond Gaussian measures to include others like Cauchy measures.

Contribution

It extends the validity of key conjectures to a broader class of rotationally invariant measures, introducing new sharp weighted Poincaré inequalities.

Findings

Confirmed B and Gardner-Zvavitch conjectures for rotationally invariant measures

Extended results to non-Gaussian measures such as Cauchy measures

Developed new sharp weighted Poincaré inequalities for these measures

Abstract

We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance to Cauchy measures as well. For the proof, new sharp weighted Poincar\'e inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.