Inequalities for mixed $p$-affine surface area

TL;DR

This paper introduces new inequalities and geometric interpretations for mixed p-affine surface areas, including a novel class of bodies called illumination surface bodies, which are not necessarily convex.

Contribution

It establishes new Alexandrov-Fenchel and affine isoperimetric inequalities for mixed p-affine surface areas and introduces illumination surface bodies with unique properties.

Findings

New inequalities for mixed p-affine surface areas.

Illumination surface bodies are not necessarily convex.

Geometric interpretations via non-convex bodies.

Abstract

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed $p$-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of $L_p$ affine surface areas, mixed $p$-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.