Construction of a Φ-function for two convex polytopes

Combinatorially Two-Orbit Convex Polytopes

Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group coincide). Hence, a combinatorially two-orbit convex polytope is isomorphic to one of a known finite list, all of which are 3-dimensional: the cuboctahedron, i...

Minimum Convex Container of Two Convex Polytopes under Translations

Given two convex d-polytopes P and Q in R for d ≥ 3, we study the problem of bundling P and Q in a smallest convex container. More precisely, our problem asks to find a minimum convex set containing P and Q that are in contact under translations. For dimension d = 3, we present the first exact algorithm that runs in O(n) time, where n denotes the number of vertices of P and Q. Our approach easi...

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...

Barycentric coordinates for convex polytopes

An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described. The key to this extension is the construction, for a given convex polytope, of a unique polynomial associated with that polytope. This polynomial, the adjoint of the polytope, generalizes a previous two-dimensional construction described by Wachspress. The barycentric coordina...

polymake: a Framework for Analyzing Convex Polytopes