Extremizers of the Alexandrov--Fenchel inequality within a new class of convex bodies

TL;DR

This paper characterizes extremizers of the Alexandrov--Fenchel inequality within a new class of convex bodies called polyoids, extending recent results and providing a geometric description of the support of associated mixed area measures.

Contribution

It introduces polyoids and characterizes them via generating measures, extending Alexandrov--Fenchel inequality extremizer results to broader convex body classes.

Findings

Characterization of polyoids using generating measures.

Extension of extremizer results to convex bodies containing polyoids and smooth bodies.

Geometric description of the support of mixed area measures.

Abstract

Mixed volumes in $n$-dimensional Euclidean space are functionals of $n$-tuples consisting of convex bodies $K,L,C_1,\ldots,C_{n-2}$. The Alexandrov--Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies, which cover as very special cases many important inequalities between basic geometric functionals. The problem of characterizing completely the equality cases in the Alexandrov--Fenchel inequality is wide open. Major recent progress was made by Yair Shenfeld and Ramon van Handel \cite{SvH22,SvH23+}, in particular they resolved the problem in the cases where $K,L$ are general convex bodies and $C_1,\ldots,C_{n-2}$ are polytopes, zonoids or smooth bodies (under some dimensional restriction). We introduce the class of polyoids, which includes polytopes, zonoids and triangle bodies, and characterize polyoids by using generating measures. Based on this characterization and Shenfeld and van Handel's contribution, we extend their result to a class of convex bodies containing all polyoids and smooth bodies. Our result is stated in terms of the support of the mixed area measure of the unit ball $B^n$ and $C_1,\ldots,C_{n-2}$. A geometric description of this support is provided in the accompanying work \cite{HugReichert23+}.