Wulff shapes and a characterization of simplices via a Bezout type inequality

TL;DR

This paper characterizes simplices via a Bezout type inequality for mixed volumes, introduces a new differentiability theorem for Wulff shapes, and explores the class of weakly decomposable convex bodies in relation to mixed volume inequalities.

Contribution

It provides a novel characterization of simplices using a Bezout type inequality and introduces a new differentiability result for Wulff shapes.

Findings

The inequality characterizes simplices among convex bodies.

A new differentiability theorem for Wulff shapes was established.

The class of weakly decomposable convex bodies was introduced and studied.

Abstract

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes $$ V(L_1,\dots,L_{n})V_n(K)\leq V(L_1,K[{n-1}])V(L_2,\dots, L_{n},K). $$ We show that the above inequality characterizes simplices, i.e. if $K$ is a convex body satisfying the inequality for all convex bodies $L_1, \dots, L_n \subset {\mathbb R}^n$, then $K$ must be an $n$-dimensional simplex. The main idea of the proof is to study perturbations given by Wulff shapes. In particular, we prove a new theorem on differentiability of the support function of the Wulff shape, which is of independent interest. In addition, we study the Bezout inequality for mixed volumes introduced in arXiv:1507.00765 . We introduce the class of weakly decomposable convex bodies which is strictly larger than the set of all polytopes that are non-simplices. We show that the Bezout inequality in arXiv:1507.00765 characterizes weakly indecomposable convex bodies.