Combinatorics Seminar

When: Sunday, April 22, 10am Where: Schreiber 309 Speaker: Imre Barany, Budapest and London Title: Recent results on random polytopes: a survey

Abstract:

A random polytope K_n is, by definition, the convex hull of n random independent, uniform points for a convex body K in R^d. The investigation of random polytopes started with Sylvester in 1864. Hundred years later R\'enyi and Sulanke began studying the expectation of various functionals of K_n, for instance number of vertices, volume, surface area, etc. Since then many papers have been devoted to deriving precise asymptotic formulae for Vol(K-K_n). But with one or two notable exceptions, very litte has been known about the distribution of this functional. In the last three years, however, two breakthrough results have been proved: Van Vu has given tail estimates for the random variables in question, and M. Reitzner has obtained a central limit theorem in the case when K is a smooth convex body. In this talk I will explain these new results and some of the subsequent development: upper and lower bounds for the variance, central limit theorems when K is a polytope, and the behaviour of Gaussian random polytopes.