Imre Bárány, Hungarian Academy of Sciences and University College London

Abstract

Given a convex cone C in R^d, an integral zonotope T is the sum of segments [0,v_i] (i=1, ... ,m) where each v_i in C is a vector with integer coordinates. The endpoint of T is k=sum_1^m v_i. Let F(C,k) be the family of all integral zonotopes in C whose endpoint is k in C. We prove that, for large k, the zonotopes in F(C,k) have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes in F(C,k) are very close to a fixed convex set which is actually a zonoid. We also establish several combinatorial properties of a typical zonotope in F(C,k). This is joint work with Julien Bureaux and Ben Lund