# Typical lattice zonotopes

## Ben Lund, University of Georgia

## March 13, 2018

A convex lattice polytope is the convex hull of a set of integral
points. Vershik conjectured that a polygon chosen uniformly at random
from the set of all 1/n scaled convex lattice polygons contained in
the unit square will be arbitrarily close to a fixed limit shape, with
arbitrarily high probability for n sufficiently large. This conjecture
is now a theorem, with three different proofs given by Bárány, by
Vershik, and by Sinai. However, the existence of a limit shape for
polytopes is open in dimensions greater than 2.

I will discuss this problem and the known results, and I will present
recent work, joint with Bárány and Bureaux, in which we show the
existence of a limit shape for lattice zonotopes in all dimensions.