  Estimating the number of vertices of a two dimensional projection, called a
shadow, of a polytope is a fundamental tool for understanding the performance
of the shadow simplex method for linear programming among other applications.
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.
