Given a polytope P⊆Rn, the Chv atal–Gomory procedure computes iteratively the integer hull PI of P. The Chv atal rank of P is the minimal number of iterations needed to obtain PI . It is always nite, but already the Chv atal rank of polytopes in R can be arbitrarily large. In this paper, we study polytopes in the 0=1 cube, which are of particular interest in combinatorial optimization. We show ...

A compressed polytope is an integral convex polytope all of whose pulling triangulations are unimodular. A (q − 1)-simplex Σ each of whose vertices is a vertex of a convex polytope P is said to be a special simplex in P if each facet of P contains exactly q − 1 of the vertices of Σ. It will be proved that there is a special simplex in a compressed polytope P if (and only if) its toric ring K[P]...

It is well known that the set of possible degree sequences for a simple graph on n vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a simple k-uniform hypergraph on n vertices is not the intersection of a lattice and a convex polytope for k > 3 and n > k + 13. We also show an analogous nonconvexity result for the set of degre...

There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic d polytope with n vertices. In this paper it is shown that both of them have the homotopy type of a sphere of dimension n d 3. Moreover, we resolve positively a new special case of the Generalized Baues Problem: The Baues poset of all polytopal decompositions of a cyclic pol...

Assume K ⊂ R is a convex body and Xn ⊂ K is a random sample of n uniform, independent points from K. The convex hull of Xn is a convex polytope Kn called random polytope inscribed in K. We are going to investigate various properties of this polytope: for instance how well it approximates K, or how many vertices and facets it has. It turns out that Kn is very close to the so called floating body...

The volume and the number of lattice points of the permutohedron Pn are given by certain multivariate polynomials that have remarkable combinatorial properties. We give 3 different formulas for these polynomials. We also study a more general class of polytopes that includes the permutohedron, the associahedron, the cyclohedron, the Stanley-Pitman polytope, and various generalized associahedra r...

The isotropy constant of any d-dimensional polytope with n vertices is bounded by C p n/d where C > 0 is a numerical constant.

The b-clique polytope CP b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes. Including the Boolean quadric polytope QP as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing...

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. Let DP(n) (respectively, DS(n)) denote the convex hull of all degree partitions (respectively, degree sequences) of simple graphs on the vertex set [n] = {1, 2, . . . , n}. We think of DS(n) as the symmetrization of DP(n) and DP(n) as the asymmetric part of DS(n). The polytope DS(n) is a well st...

We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN(f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler variety...