Random Shadows of Fixed Polytopes

Abstract Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool appearing in several applications through connections to linear programming and parametric optimization. We prove multiple upper bounds on the expected number of vertices of a random shadow of a fixed polytope. Our bounds are in terms of various parameters in the literature including geometric diameter and edge lengths, minimal and maximal slack, maximal coordinates for lattice polytopes, and maximum absolute values of subdeterminants. For the case of geometric diameter and edge lengths, we prove lower bounds and argue that our upper and lower bounds are both tight for zonotopes.