Extremal Properties of 0/1-Polytopes

Abstract

Abstract. We provide lower and upper bounds for the maximal number of facets of a d-dimensional 0/1-polytope, and for the maximal number of vertices that can appear in a two-dimensional projection ("shadow") of such a polytope.

Simple 0/1-Polytopes

“…It is well known (see, for example, [4]) that if P and Q as above are polar to P and Q , respectively, then P ⊕ Q is polar to P × Q . Thus in order to prove the corollary, we shall show how to realize free sums of 0/1-simplices (that are, of course, polar to itself) as 0/1polytopes.…”

“…Unfortunately, we cannot apply the free sum construction of [4] directly because there is no two-dimensional 0/1-simplex which contains the point (1/2, 1/2) in its interior. Instead we use the following slight modification of the construction of [4].…”

“…However, only very few properties of general 0/1-polytopes have been studied to date. Examples for results into this direction are the fact that the diameter of a ddimensional 0/1-polytope may not be larger than d [5], or estimates on the maximal number of facets that a d-dimensional 0/1-polytope can have [3,4], as well as on the sizes of coefficients of inequalities describing facets of 0/1-polytopes (following from [1]). …”

Upper Bounds on the Maximal Number of Facets of 0/1-Polytopes

“…On the other hand, Bárány (see [11]) gave a good argument that a d-dimensional 0/1-polytope cannot have more than d!+2d facets, which we will briefly review below (Lemma 2) since we will need it in one of our proofs. Let f (d) be the maximal number of facets that a d-dimensional 0/1-polytope can have.…”

“…The d-dimensional cross-polytope can be realized (combinatorially) as the 0/1-polytope conv{e i , 1 − e i : 1 ≤ i ≤ d}, where e i is the ith canonical unit vector and 1 is the all-ones vector, showing that d-dimensional 0/1-polytopes can have as many as 2 d facets. Starting with a special randomly generated 0/1-polytope of dimension 13 with more than 17 million facets (found by Christof [7]), and using some inductive construction due to Kortenkamp et al [11], one can show that the maximal numbers of facets of d-dimensional 0/1-polytopes grow at least as fast as 3.6 d .…”