The set of bistochastic or doubly stochastic N × N matrices form a convex set called Birkhoff’s polytope, that we describe in some detail. Our problem is to characterize the set of unistochastic matrices as a subset of Birkhoff’s polytope. For N = 3 we present fairly complete results. For N = 4 partial results are obtained. An interesting difference between the two cases is that there is a ball...

The optimal k-restricted 2-factor problem consists of nding, in a complete undirected graph K n , a minimum cost 2-factor (subgraph having degree 2 at every node) with all components having more than k nodes. The problem is a relaxation of the well-known symmetric travelling salesman problem, and is equivalent to it when n 2 k n ? 1. We study the k-restricted 2-factor polytope. We present a lar...

This lecture describes a data structure for representing convex polytopes and a divide and conquer algorithm for computing convex hull in 3 dimensions. Let S be a set of n points in . Convex hull of S (CH(S)) is the smallest convex polytope that contains all n points. Since the boundary of this polytope is planar, it can be efficiently represented by the data structure described in the next sec...

Let P be a convex polytope in R, d = 3 or 2, with n vertices. We present linear time algorithms for approximating P by simpler polytopes. For instance, one such algorithm selects k < n vertices of P whose convex hull is the approximating polytope. The rate of approximation, in the Hausdorff distance sense, is best possible in the worst case. An analogous algorithm, where the role of vertices is...

For a given convex body K in Rd, a random polytope K(n) is defined (essentially) as the intersection of n independent closed halfspaces containing K and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of K(n) and K as n tends to infinity. For a simplicial polytope P, a precise asymptotic...

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 varie...

We prove the central limit theorem for the volume and the f-vector of the Poisson random polytope η in a fixed convex polytope P ⊂ R d. Here, η is the convex hull of the intersection of a Poisson process X of intensity η with P. 1. Introduction and main results. Let K ⊂ R d be a convex set of volume 1. Assume that X = X(η) is a Poisson point process in R d of intensity η. The intersection of K ...

The Birkhoff polytope Bn is the convex hull of all (n×n) permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics. In this paper we study combinatorial types L of faces of a Birkhoff polytope. The Birkhoff dimension bd(L) of L is the smallest n s...

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 varie...

We study facets of the k-partition polytope Pk.+, the convex hull of edges cut by r-partitions of a complete graph for r < k, k 2 3. We generalize the hypermetric and cycle inequalities (see Deza and Laurent, 1992) from the cut polytope to Pk.,, k 2 3. We give some sufficient conditions under which these are facet defining. We show the anti-web inequality introduced by Deza and Laurent (1992) t...