On the analogue of the concavity of entropy power in the Brunn-Minkowski theory

TL;DR

This paper explores the conjectured analogy between the concavity of entropy power and the Brunn-Minkowski inequality, examining its validity and connections to geometric inequalities.

Contribution

It investigates Costa and Cover's conjecture on the $rac{1}{n}$-concavity of outer parallel volume as an analogue of entropy power concavity.

Findings

Provides evidence supporting the conjecture

Establishes links between entropy power inequality and geometric inequalities

Offers new insights into the geometric interpretation of entropy

Abstract

Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured in {\it On the similarity of the entropy power inequality and the Brunn-Minkowski inequality} (IEEE Trans. Inform. Theory 30 (1984), no. 6, 837-839) the $\frac{1}{n}$-concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We investigate this conjecture and study its relationship with geometric inequalities.