The support of mixed area measures involving a new class of convex bodies

TL;DR

This paper characterizes the support of mixed area measures for polyoids and smooth bodies, extending previous results and confirming a long-standing conjecture in convex geometry.

Contribution

It provides a geometric description of the support of mixed area measures for polyoids, advancing the understanding of equality cases in the Alexandrov--Fenchel inequality.

Findings

Support of mixed area measure characterized for polyoids and smooth bodies

Extension of previous results to a broader class of convex bodies

Confirmation of Schneider's long-standing conjecture for polyoids

Abstract

Mixed volumes in $n$-dimensional Euclidean space are functionals of $n$-tuples of convex bodies $K,L,C_1,\ldots,C_{n-2}$. The Alexandrov--Fenchel inequalities are fundamental inequalities between mixed volumes of convex bodies. As very special cases they cover or imply many important inequalities between basic geometric functionals. A complete characterization of the equality cases in the Alexandrov--Fenchel inequality remains a challenging open problem. 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 $C_1,\ldots,C_{n-2}$ are polytopes, zonoids or smooth bodies (under some dimensional restriction). In \cite{HugReichert23+} we introduced the class of polyoids, which are defined as limits of finite Minkowski sums of polytopes having a bounded number vertices. Polyoids encompass polytopes, zonoids and triangle bodies, and they can be characterized by means of generating measures. Based on this characterization and Shenfeld and van Handel's contribution, we extended their result to polyoids (or smooth bodies). Our previous result was stated in terms of the support of the mixed area measure associated with the unit ball $B^n$ and $C_1,\ldots,C_{n-2}$. This characterization result is completed in the present work which more generally provides a geometric description of the support of the mixed area measure of an arbitrary $(n-1)$-tuple of polyoids (or smooth bodies). The result confirms a long-standing conjecture by Rolf Schneider in the case of polyoids and hence, in particular, of zonoids.