Segmentos al azar en Eⁿ

Abstract

Let En be the Euclidean space of n dimensions. We consider sets of line segments given at random in En with the origin and direction uniformly distributed and with a given length distribution dF (l) (0 < l < [infinit]). We first consider the problem of finding the probability that a segment which intersects a given convex body Q have 0, 1 or 2 common points with the boundary of Q. Then we consider a Poisson process of segments of intensity [lamda] and solve some problems, in particular, that of finding the Distribution of the distance from a fixed point, chosen independently of the process, to the nearest end of the line segments. For n = 1, 2, 3 this last problem has been solved by R. Coleman. Finally, we add some comments on random trees in En