We introduce the interval order polytope of a digraph D as the convex hull of interval order inducing arc subsets of D. Two general schemes for producing valid inequalities are presented. These schemes have been used implicitly for several polytopes and they are applied here to the interval order polytope. It is shown that almost all known classes of valid inequalities of the linear ordering po...

A theorem of Howe states that every 3-dimensional lattice polytope P whose only lattice points are its vertices, is a Cayley polytope, i.e. P is the convex hull of two lattice polygons with distance one. We want to generalize this result by classifying 3-dimensional lattice polytopes without interior lattice points. The main result will be, that they are up to finite many exceptions either Cayl...

Let P be a convex polytope containing the origin, whose dual is a lattice polytope. Hibi’s Palindromic Theorem tells us that if P is also a lattice polytope then the Ehrhart δ-vector of P is palindromic. Perhaps less well-known is that a similar result holds when P is rational. We present an elementary lattice-point proof of this fact.

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a gener...

In this paper we study some geometrical questions about the polytope of bi-capacities. For this, introduce concept pointed order polytope, a natural generalization polytopes. Basically, is that takes advantage relation partially ordered set and such there relevant element in structure. We which are vertices polytopes sort out simple way to determine whether two adjacent. also general form its f...

A compressed polytope is an integral convex polytope any of whose reverse lexicographic initial ideals is squarefree. A sufficient condition for a (0, 1)-polytope to be compressed will be presented. One of its immediate consequences is that the class of compressed (0, 1)-polytopes includes (i) hypersimplices, (ii) order polytopes of finite partially ordered sets, and (iii) stable polytopes of p...

A positroid is a special case of realizable matroid that arose from the study totally nonnegative part Grassmannian by Postnikov. In this paper, we facets its polytope and independent set polytope. This allows one to describe bases sets directly decorated permutation, bypassing use Grassmann necklace. We also criterion for determining whether given cyclic interval flat or not using then show ho...

We investigate the question of whether any d-colorable simplicial d-polytope can be octahedralized, i.e., subdivided to a d-dimensional geometric cross-polytopal complex. give positive answer in dimension 3, with additional property that octahedralization introduces no new vertices on boundary polytope.

Epitope-based vaccination strategies designed to induce tumor-specific CD8 CTL are being widely considered for cancer immunotherapy. Here we describe a recombinant poxvirus vaccine that codes for ten HLA-A2-restricted epitopes derived from five melanoma Ags conjoined in an artificial polyepitope or polytope construct. Target cells infected with the melanoma polytope vaccinia were recognized by ...

A (convex) polytope is the convex hull of finitely many points in Rd. Alternatively, a polytope can be defined as the bounded intersection of finitely many halfspaces, that is, a polytope is the (bounded) solution set of a system of linear equations and inequalities. (This latter fact is the main non-geometric reason why people are interested in polytopes—linear systems are everywhere.) The mai...