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A conjecture of Mihail and Vazirani [5] states that the edge expansion of the graph of every 0/1 polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a 0/1 polytope in R d is greater than 1 over some polynomial function of d. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random 0/1 polytope in R d is at least 1 12d with high probability.

Gaussian random polytopes have received a lot of attention especially in the case where the dimension is fixed and the number of points goes to infinity. Our focus is on the less studied case where the dimension goes to infinity and the number of points is proportional to the dimension d. We study several natural quantities associated to Gaussian random polytopes in this setting. First, we show that the expected number of facets is equal to C(α) d+o(d) where C(α) is some constant which depends on the constant of proportionality α. We also extend this result to the expected number of k-facets. We then consider the more difficult problem of the asymptotics of the expected number of pairs of estranged facets of a Gaussian random polytope. When n = 2d, we determined the constant C so that the expected number of pairs of estranged facets is equal to C d+o (d) .