Entropic versions of Bergstr\"om's and Bonnesen's inequalities

TL;DR

This paper develops entropic analogues of classical geometric inequalities, strengthening existing inequalities for Fisher information and entropy power in higher dimensions, and characterizes equality cases.

Contribution

It introduces novel entropic versions of Bergström's and Bonnesen's inequalities, extending their applicability to entropy power and Fisher information.

Findings

Strengthens convolution inequality for Fisher information.

Generalizes entropy power inequality to higher dimensions.

Characterizes equality cases in entropic Bonnesen inequality.

Abstract

We establish analogues of the Bergstr\"om and Bonnesen inequalities, related to determinants and volumes respectively, for the entropy power and for the Fisher information. The obtained inequalities strengthen the well-known convolution inequality for the Fisher information as well as the entropy power inequality in dimensions $d>1$, while they reduce to the former in $d=1$. Our results recover the original Bergstr\"om inequality and generalize a proof of Bergstr\"om's inequality given by Dembo, Cover and Thomas. We characterize the equality case in our entropic Bonnesen inequality.