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 ( L ₁ ,..., L _n ) V _n ( K ) ≤ V ( L ₁ , K [n-1]) V ( L ₂ ,..., 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 , ..., L _n ⊂ 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.