Then Ehrhart’s theorem asserts that p(N, f) is a polynomial in N . In the case that ∆ is a simple polytope (meaning that n edges emanate from each vertex) Ehrhart’s theorem is a consequence of the Euler-MacLaurin formula, [Kh, KP, CS1, CS2, Gu, BV, DR] and one can be more explicit about the nature of the polynomial p(N, f). Let us explain how this works in the more restrictive case where ∆ is n...

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n− d. That is to say, we can go from any vertex to any other vertex using at most n− d edges. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite s...

The extension complexity of a polytope P is the smallest integer k such that P is the projection of a polytope Q with k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(logn), a result originating from work by Ben-Tal and Nemirovski (Math. Oper. Res. 26(2), 193–205, 2001). Our proof easily generaliz...

In this paper we explore the combinatorial automorphism group of the linear ordering polytope P n LO for each n > 1. We establish that this group is isomorphic to Z 2 Sym(n + 1) if n > 2 (and to Z 2 if n = 2). Doing so, we provide a simple and uniied interpretation of all the automorphisms.

The type An root polytope P(An ) is the convex hull in R of the origin and the points ei − ej for 1 ≤ i < j ≤ n + 1. Given a tree T on vertex set [n + 1], the associated root polytope P(T ) is the intersection of P(An ) with the cone generated by the vectors ei − ej , where (i, j) ∈ E(T ), i < j. The reduced forms of a certain monomial m[T ] in commuting variables xij under the reduction xijxjk...

We study a natural generalization of the noncrossing relation between pairs of elements in [n] to k-tuples in [n] that was first considered by Petersen et al. [J. Algebra 324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on ([n] k ) induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset ...

We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertexand halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction,...

An object representation scheme which allow_ fast Eucltdea,7 distance cal,:vlatu_rz is prese_ted. The object model eztends the polytope model by representLny objects as the convez hull of a finite set of spheres. An algorithm for calculating distances between objects is developed which is linear in the total number of spheres specifying the two objects.