The polytope Q, of the convex combinations of the permutation matrices of order n is well known (Birkhoff’s theorem) to be the polytope of doubly stochastic matrices of order n. In particular it is easy to decide whether a matrix of order n belongs to Q,. . check to see that the entries are nonnegative and that all row and columns sums equal 1. Now the permutations z of { 1, 2, . . . . n} are i...

Given a probability assignement to n events A1,...,An in a finite probability space we show that the set of all coherent probabilities on the algebra A generated by A1,...,An is a polytope, that we call the coherent probability polytope and we describe explicitely. This polytope structure implies in particular the "fundamental theorem of probability" of De Finetti [4], which states that for eve...

Consider the moment curve in the real Euclidean space R defined parametrically by the map γ : R → R, t → γ(t) = (t, t, . . . , t). The cyclic d-polytope Cd(t1, . . . , tn) is the convex hull of the n, n > d, different points on this curve. The matroidal analogues are the alternating oriented uniform matroids. A polytope [resp. matroid polytope] is called cyclic if its face lattice is isomorphic...

We present explicit constructions of centrally symmetric polytopes with many faces: (1) we construct a d-dimensional centrally symmetric polytope P with about 3d/4 ≈ (1.316)d vertices such that every pair of non-antipodal vertices of P spans an edge of P , (2) for an integer k ≥ 2, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitra...

Let m,n ≥ 1 be integers. Define Tm,n to be the transportation polytope consisting of the m × n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case Bn = Tn,n is the much-studied Birkhoffvon Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determ...

we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ of a finite dimensional representation $v$ of a connected reductive group. we describe the cone generated by $s$ as the cone over the weight polytope of $v$ intersected with the positive weyl chamber. from this we get a description for the asymptotic of the number of highest weights appearing in $v^{otime...

A perfect matching in an undirected graph G = (V,E) is a set of vertex disjoint edges from E that include all vertices in V . The perfect matching problem is to decide if G has such a matching. Recently Rothvoß proved the striking result that the Edmonds’ matching polytope has exponential extension complexity. Here we describe a perfect matching polytope that is different from Edmonds’ polytope...

A compact formulation of the set of tours neither in a graph nor its complement is presented and illustrates a general methodology proposed for constructing polyhedral models of decision problems based upon permutations, projection and lifting techniques. Directed Hamilton tours on n vertex graphs are interpreted as (n-1)- permutations. Sets of extrema of Birkhoff polyhedra are mapped to tours ...

We show that every heptagon is a section of a 3-polytope with 6 vertices. This implies that every n-gon with n > 7 can be obtained as a section of a (2 + ⌊ n 7 ⌋ )dimensional polytope with at most ⌈ 6n 7 ⌉ vertices; and provides a geometric proof of the fact that every nonnegative n ×m matrix of rank 3 has nonnegative rank not larger than ⌈ 6min(n,m) 7 ⌉ . This result has been independently pro...