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 f(d, n) of facets and the minimum average degree a(d, n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy f(d, n) <= 3d and a(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.Comment: Accepted for publication in J. Comb. Theory Ser. A; 24 pages; simplified proof of Theorem 1; corrected and improved version of Theorem 4 (the average degree is now bounded by d+4 instead of d+8); several minor corrections suggested by the referee

Affine hom-complexes

For two general polytopal complexes the set of face-wise affine maps between them is shown to be a polytopal complex in an algorithmic way. The resulting algorithm for the affine hom-complex is analyzed in detail. There is also a natural tensor product of polytopal complexes, which is the left adjoint functor for Hom. This extends the corresponding facts from single polytopes, systematic study of which was initiated in [6,12]. Explicit examples of computations of the resulting structures are included. In the special case of simplicial complexes, the affine hom-complex is a functorial subcomplex of Kozlov's combinatorial hom-complex [14], which generalizes Lovasz' well-known construction [15] for graphs.Comment: final version, to appear in Portugaliae Mathematic

Convex normality of rational polytopes with long edges

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope P has lattice length at least 4d(d+1) then P is normal. This answers in the positive a question raised in 2007. If P is a lattice simplex whose edges have lattice lengths at least d(d+1) then P is even covered by lattice parallelepipeds. For the approach developed here, it is necessary to involve rational polytopes even for applications to lattice polytopes.Comment: 16 pages, final version, to appear in Advances in Mathematic