Perturbation of central transportation polytopes of order kn× n
نویسنده
چکیده
We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order kn × n, we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order kn× n. Résumé. Nous décrivons une méthode de perturbation qui peut être utilisé pour calculer la fonction génératrice multivariée (MGF) d’un polyèdre non-simple, et ensuite construire une perturbation qui fonctionne pour tout polytope de transport. Appliquant cette perturbation à la famille des centraux de transport polytopes de l’ordre kn × n, nous obtenons des formules pour le MGF du polytope. Les formules que nous obtenons sont énumérées par les objets combinatoires. Un cas spécial des formules récupère les résultats sur des polytopes de Birkhoff donnés par l’auteur et De Loera et Yoshida. Nous récupérons également la formule pour le nombre de sommets maximum des de transport polytopes d’ordre kn× n.
منابع مشابه
On Mean Outer Radii of Random Polytopes
In this paper we introduce a new sequence of quantities for random polytopes. Let KN = conv{X1, . . . ,XN} be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body K of R. We prove that the so-called k-th mean outer radius R̃k(KN ) has order max{ √ k, √ logN}LK with high probability if n ≤ N ≤ e √ . We also show that this is also the right or...
متن کاملOn the variance of random polytopes
A random polytope is the convex hull of uniformly distributed random points in a convex body K. A general lower bound on the variance of the volume and f -vector of random polytopes is proved. Also an upper bound in the case when K is a polytope is given. For polytopes, as for smooth convex bodies, the upper and lower bounds are of the same order of magnitude. The results imply a law of large n...
متن کاملCentral Limit Theorems for Gaussian Polytopes
Choose n random, independent points in R according to the standard normal distribution. Their convex hull Kn is the Gaussian random polytope. We prove that the volume and the number of faces of Kn satisfy the central limit theorem, settling a well known conjecture in the field.
متن کاملCentral Limit Theorems for Random Polytopes in a Smooth Convex Set
Let K be a smooth convex set with volume one in R. Choose n random points in K independently according to the uniform distribution. The convex hull of these points, denoted by Kn, is called a random polytope. We prove that several key functionals of Kn satisfy the central limit theorem as n tends to infinity.
متن کاملVolume Thresholds for Gaussian and Spherical Random Polytopes and Their Duals
Let g be a Gaussian random vector in R. Let N = N(n) be a positive integer and let KN be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes VN := E vol(KN ∩RB 2 )/ vol(RB 2 ). For a large range of R = R(n), we establish a sharp threshold for N , above which VN → 1 as n → ∞, and below which VN → 0 as n → ∞. We also consider the case when KN is generated by ...
متن کامل