Analysis and Probability Seminar: “Wulff Shapes and a Characterization of Simplices via a Bezout Type Inequality”

Tuesday April 17, 2018 3:00pm Yost 306


Speaker: Christos Saraoglu, Kent State University

Title: Wulff Shapes and a Characterization of Simplices Via a Bezout Type Inequality

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 \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 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¬†characterizes weakly indecomposable convex bodies.”

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