Combinatorial face enumeration in convex polytopes

Combinatorial Aspects of Convex Polytopes

Some Aspects of the Combinatorial Theory of Convex Polytopes

We start with a theorem of Perles on the k-skeleton, Skel k (P) (faces of dimension k) of d-polytopes P with d+b vertices for large d. The theorem says that for xed b and d, if d is suuciently large, then Skel k (P) is the k-skeleton of a pyramid over a (d ? 1)-dimensional polytope. Therefore the number of combinatorially distinct k-skeleta of d-polytopes with d + b vertices is bounded by a fun...

From Polytopes to Enumeration

In the problem session you saw that v-e+f=2 for 3-polytopes. Is something similar true in higher dimensions? Are there other restrictions on the number of faces of each dimension? What is the most number of faces in each dimension if we fix the number of vertices and dimension? The least? What if we fix other numbers of faces in random dimensions? Notice that all of these questions involve coun...

Convex Polytopes

The study of convex polytopes in Euclidean space of two and three dimensions is one of the oldest branches of mathematics. Yet many of the more interesting properties of polytopes have been discovered comparatively recently, and are still unknown to the majority of mathematicians. In this paper we shall survey the subject, mentioning some of the most recent results, and stating the more importa...

Enumeration of 2-Level Polytopes

A (convex) polytope P is said to be 2-level if for every direction of hyperplanes which is facet-defining for P , the vertices of P can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial t...