Poisson flats in Euclidean spaces Part I: A finite number of random uniform flats

Abstract

This is the first part of a three part work, the common setting being E^d, Euclidean space of d dimensions. Many random physical phenomena, often in the form of structures, admit models which are assemblages of random s-flats in E^d. Indeed, taking s = 0, any n-sample from a d-dimensional distribution may of course be so regarded! For static phenomena generally 0 ≦ s < d ≦ 3, while 4 is to be substituted for 3 if time variation is allowed. By postulating a high degree of stochastic independence and uniformity, a variety of “simple” models is defined. However, their investigation poses problems in geometrical probability of a wide range of difficulty; the solutions of many of which are still far from complete.