Extremal general affine surface areas

TL;DR

This paper introduces extremal general affine surface areas for convex bodies, proves the existence of extremal bodies, and establishes inequalities analogous to Blaschke-Santaló, extending recent $L_p$ affine surface area research.

Contribution

It defines and analyzes extremal affine surface areas, proves their extremal bodies exist, and establishes new inequalities, extending the $L_p$ affine surface area framework to an Orlicz setting.

Findings

Existence of extremal convex bodies for the defined functionals.

Continuity of the extremal affine surface area functionals.

Blaschke-Santaló and inverse inequalities for extremal affine surface areas.

Abstract

For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal general affine surface areas, defined by \[ {\rm IS}_{\varphi}(K):=\sup_{K^\prime\subset K}{\rm as}_{\varphi}(K),\quad {\rm os}_{\psi}(K):=\inf_{K^\prime\supset K}{\rm as}_{\psi}(K) \] where ${\rm as}_{\varphi}(K)$ and ${\rm as}_{\psi}(K)$ are the $L_\varphi$ and $L_\psi$ affine surface area of $K$, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ${\rm IS}_{\varphi}$ and ${\rm os}_{\psi}$ are continuous. In our main results, we prove Blaschke-Santal\'o type inequalities and inverse Santal\'o type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Sch\"utt and Werner (2020), who introduced and studied the extremal $L_p$ affine surface areas.