Revlex-initial 0/1-polytopes

We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers gnfac(d,n) of facets and the minimum average degree gavdeg(d,n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy gnfac(d,n)⩽3d and gavdeg(d,n)⩽d+4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.