To logconcavity and beyond

TL;DR

This paper explores the preservation of a new, stronger form of concavity by the heat flow, extending the classical result that it preserves logconcavity, and identifies the most robust concavity property maintained.

Contribution

It introduces a variation of concavity stronger than logconcavity and determines the strongest such property preserved by the heat flow.

Findings

Identifies a stronger concavity property preserved by heat flow.

Shows heat flow preserves this new concavity.

Determines the maximal class of concavity preserved.

Abstract

In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow.