Aproximación aleatoria de cuerpos convexos

Abstract

Problems related to the random approximation of convex bodies fall into the field of integral geometry and geometric probabilities. The aim of this paper is to give a survey of known results about the stochastic model that has received special attention in the literature and that can be described as follows: Let K be a d-dimensional convex body in Eucliden space Rd, d = 2. Denote by Hn the convex hull of n independent random points X1, ..., Xn distributed identically and uniformly in the interior of K. If f is a random variable on d-dimensional polytopes on Rd, we define the random variable fn by: fn = f (conv {X1, ..., Xn}), where conv denotes the convex hull. Typical random variables studied in the literature are numbers of vertices and facets, volume, surface area and mean width. Our main interest concerns the study of the mathematical expectation E(fn) of fn. Some further stochastic models and other problems related to random points studied in the literature will be presented.