The number of polytopes, configurations and real matroids

The Number of Polytopes, Configurations and Real Matroids

We show that the number of combinatorially distinct labelled d-polytopes on n vertices is at most (n/oo. A similar bound for the number of simplicial polytopes has previously been proved by Goodman and Pollack. This bound improves considerably the previous known bounds. We also obtain sharp upper and lower bounds for the numbers of real oriented and unoriented matroids with n elem...

Cyclic Polytopes and Oriented Matroids

Consider the moment curve in the real Euclidean space R defined parametrically by the map γ : R → R, t → γ(t) = (t, t, . . . , t). The cyclic d-polytope Cd(t1, . . . , tn) is the convex hull of the n, n > d, different points on this curve. The matroidal analogues are the alternating oriented uniform matroids. A polytope [resp. matroid polytope] is called cyclic if its face lattice is isomorphic...

Subgraph polytopes and independence polytopes of count matroids

Given an undirected graph, the non-empty subgraph polytope is the convex hull of the characteristic vectors of pairs (F, S) where S is a nonempty subset of nodes and F is a subset of the edges with both endnodes in S. We obtain a strong relationship between the non-empty subgraph polytope and the spanning forest polytope. We further show that these polytopes provide polynomial size extended for...

Constructing neighborly polytopes and oriented matroids

A d-polytope P is neighborly if every subset of b d 2 c vertices is a face of P . In 1982, Shemer introduced a sewing construction that allows to add a vertex to a neighborly polytope in such a way as to obtain a new neighborly polytope. With this, he constructed superexponentially many different neighborly polytopes. The concept of neighborliness extends naturally to oriented matroids. Duals o...

Matroids: h-vectors, Zonotopes, and Lawrence Polytopes