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...

Universal polytope is the polytope de ned as the convex hull of the characteristic vectors of all triangulations for a given point con guration. The equality system de ning this polytope was found, but the system of inequalities are not known yet. Larger polytopes, corresponding to linear programming relaxations, have been used in practice. We show that (1) the universal polytope, the polytope ...

The Birkhoff polytope Bn is the convex hull of all n× n permutation matrices in Rn×n. We compute the combinatorial symmetry group of the Birkhoff polytope. A representation polytope is the convex hull of some finite matrix group G 6 GL(d,R). We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the face lattice of ...

Abstract We survey the computation of polytope volumes by algorithms Normaliz to which Lawrence algorithm has recently been added. It enabled us master volume computations for polytopes from social choice in dimension 119. This challenge required a sophisticated implementation algorithm.

We build facets of the quadratic 0-1 knapsack polytope following two different approaches. The quadratic 0-1 knapsack polytope is included in the Boolean quadric polytope introduced by Padberg [12] for unconstrained 0-1 quadratic problem. So in a first approach, we ask the question which are the facets of the Boolean quadric polytope that are still facets of the quadratic 0-1 knapsack polytope....

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is eq...