On local Liakopoulos-Meyer type inequalities and their functional counterparts

TL;DR

This paper establishes new local inequalities for convex bodies and log-concave functions, extending classical results and solving open questions in convex geometry and functional analysis.

Contribution

It introduces a functional Rogers-Shephard type inequality for log-concave functions and derives sharp local Liakopoulos-Meyer inequalities for convex bodies, addressing previously open problems.

Findings

Established a functional Rogers-Shephard inequality for log-concave functions.

Derived sharp local Liakopoulos-Meyer inequalities for convex bodies.

Solved open questions posed by previous researchers in the field.

Abstract

We provide a functional Rogers-Shephard type inequality for log-concave functions on $\mathbb R^n$ and any $1$-reducible $s$-cover of $[n]$. As a consequence, we derive a sharp local Liakopoulos-Meyer type inequality for $n$-dimensional convex bodies and $1$-reducible $s$-covers of any $\sigma\subset[n]$, solving a question studied by Brazitikos, Giannopoulos, Liakopoulos in [14] as well as Alonso-Guti\'errez, Bernu\'es, Brazitikos, Carbery in [3].