The Lp Polar bodies of shadow system and related inequalities

TL;DR

This paper extends the theory of $L_p$ support functions and polar bodies to shadow systems, proving convexity and concavity properties, and establishing new inequalities including a reverse Rogers-Shephard type inequality.

Contribution

It introduces convexity and concavity results for $L_p$ support functions and polar bodies within shadow systems, along with related inequalities and a reverse Rogers-Shephard inequality.

Findings

Convexity of $L_p$-support function of shadow systems.

Concavity of volume of $L_p$ polar bodies' sections.

Establishment of a reverse Rogers-Shephard inequality.

Abstract

The $L_p$ versions of the support function and polar body are introduced by Berndtsson, Mastrantonis and Rubinstein in \cite{Berndtsson-Mastrantonis-Rubinstein-2023} recently. In this paper, we prove that the $L_p$-support function of the shadow system $K_t$ introduced by Rogers and Shephard in \cite{rogers-1958-02,shephard-1964} is convex and the volume of the section of $L_p$ polar bodies of $K_t$ is $\frac{1}{n}$-concave with respect to parameter $t$, and obtain some related inequalities. Finally, we present the reverse Rogers-Shephard type inequality for $L_p$-polar bodies.