Extemizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes

TL;DR

This paper explores necessary boundary conditions for convex bodies to satisfy certain mixed volume inequalities related to the Bezout inequalities, providing a new solution for the three-dimensional case.

Contribution

It identifies boundary structure conditions for convex bodies satisfying Bezout inequalities and solves the 3D case completely.

Findings

Derived necessary boundary conditions for convex bodies

Proved the conjecture for the 3-dimensional case

Extended understanding of inequalities characterizing simplices

Abstract

In [SZ], Soprunov and Zvavitch have translated the Bezout inequalities (from Algebraic Geometry) into inequalities of mixed volumes satisfied by the simplex. They conjecture this set of inequalities characterizes the simplex, among all convex bodies in R^n. Together with Saroglou, they proved the characterization among all polytopes [SSZ1] and, for a larger set of inequalities, among all convex bodies [SSZ2]. The conjecture remains open for n \geq 4. In this work, we investigate necessary conditions on the structure of the boundary of a convex body K, for K to satisfy all inequalities. In particular, we obtain a new solution of the 3-dimensional case.