Let G be a discrete group generated by reflections in hyperbolic or Euclidean space, and H ⊂ G be a finite index subgroup generated by reflections. Suppose that the fundamental chamber of G is a finite volume polytope with k facets. In this paper, we prove that the fundamental chamber of H has at least k facets. 1. Let X be hyperbolic space H, Euclidean space E or spherical space S . A polytope...

We analyze the facial structure of the polytope associated to the GMSTP with the aim of nding \good" inequalities to strengthen our previous linear formulations. Several families of inequalities which are facet-inducing for the polytope of the GMSTP are investigated. Our proofs of \facetness" for valid inequalities of the GMSTP polytope use tools developped in [3] and also classical results fro...

Many problems in optimization lead to a cost function in the form of an integral over a polytope. To find the gradient or Hessian of such costs one needs to take the derivative of an integral over a convex polytope with respect to the parameters defining the polytope. The contribution of the current paper is to present a formula for such derivatives.

A parallelotope is a polytope whose translation copies fill space without gaps and intersections by interior points. Voronoi conjectured that each parallelotope is an affine image of the Dirichlet domain of a lattice, which is a Voronoi polytope. We give several properties of a parallelotope and prove that each of them is equivalent to it is an affine image of a Voronoi polytope.

We prove that any finite, abstract n-polytope is covered by a finite, abstract regular n-polytope.

Richard Stanley introduced the order polytope O(P ) and the chain polytope C(P ) arising from a finite partially ordered set P , and showed that the Ehrhart polynomial of O(P ) is equal to that of C(P ). In addition, the unimodular equivalence problem of O(P ) and C(P ) was studied by the first author and Nan Li. In the present paper, three integral convex polytopes Γ(O(P ),−O(Q)), Γ(O(P ),−C(Q...

Using sutured Floer homology (in short SFH) I will define a polytope inside the second relative cohomology group of a sutured manifold. This is a generalization of the dual Thurston norm polytope of a link-complement studied by Ozsvath and Szabo using link Floer homology. The polytope is maximal dimensional under certain conditions. Moreover, surface decompositions correspond to the faces of th...