The shape of any facet of a 3-polytope can be preassigned. That is, given any 3-polytope P a facet F of P and a polygon F’ with the same number of sides as F, there is an isomorphic 3-polytope P’ such that F is a facet of P’ and the isomorphism takes F onto F (see [ 21). It is also known that given any simple circuit C in the graph of a 3-polytope P, there is an isomorphic 3-polytope P’ such th...

We study a natural generalization of the noncrossing relation between pairs of elements in [n] to k-tuples in [n]. We show 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 given by the product [k]× [n− k] of two chains, and it is the join of a simplex and a sphere (that is, it is a Gorenstei...

We provide an explicit combinatorial formula for the volume of the polytope of n× n doubly-stochastic matrices, also known as the Birkhoff polytope. We do this through the description of a generating function for all the lattice points of the closely related polytope of n × n real non-negative matrices with all row and column sums equal to an integer t. We can in fact recover similar formulas f...

Deene transportation polytope Tn;m to be a polytope of nonnegative n m matrices with row sums equal to m and column sums equal to n. We present an eecient algorithm for computing the numbers f k of the k-dimensional faces for the transportation polytope T n;n+1. The construction relies on the new recurrence relation for the numbers f i , which is of independent interest.

An abstract polytope is flat if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for n > 8, a chiral n-polytope has at least 48(n− 2)(n− 2)! flags.

The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n−d. That is, any two vertices of the polytope can be connected by a path of at most n− d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope wit...

Define the transportation polytope Tn,m to be a polytope of non-negative n×m matrices with row sums equal to m and column sums equal to n. We present a new recurrence relation for the numbers fk of the k-dimensional faces for the transportation polytope Tn,n+1. This gives an efficient algorithm for computing the numbers fk , which solves the problem known to be computationally hard in a general...

The n Birkhoff polytope is the set of all doubly stochastic n × n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these polytopes, which have been known for n ≤ 8. We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function coun...

We consider the relaxation of the matching polytope defined by the non-negativity and degree constraints. We prove that given an undirected graph on n nodes and the corresponding relaxation of the matching polytope, n /2 iterations of the Lovász-Schrijver semidefinite lifting procedure are needed to obtain the matching polytope, in the worst case. We show that n /2 iterations of the procedu...