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

Inspired by a fundamental theorem of Bernstein, Kushnirenko, and Khovanskii we study the following Bezout type inequality for mixed volumes V(L1,…,Ln)Vn(K)≤V(L1,K[n−1])V(L2,…,Ln,K). We show that the above inequality characterizes simplices, i.e. if K is a convex body satisfying the inequality for all convex bodies L1,…,Ln⊂Rn, 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 [32]. 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 any weakly decomposable convex body fails to satisfy the Bezout inequality.