The mean breadth of a random polytope in a convex body

Abstract

Since the papers of R6nyi and Sulanke [-4, 5, 6] several authors have studied the mean values of geometrical functionals of the convex hull of independently and identically distributed random points in a convex body K with interior points in d-dimensional Euclidean space IR d or of a convex polytope generated by n i.i.d, random hyperplanes, e.g. Efron [2], Schmidt [7], Carnal [1], Ziezold [10], Pr6kopa [-3], Sulanke and Wintgen [9]. Schneider and Wieacker [8] generalize a result of R6nyi and Sulanke [5] on the asymptotic behaviour of the expectation of the perimeter of the convex hull of N independent random points in K for the case that K has a sufficiently smooth boundary OK. They use the fact that for d = 2 the perimeter L of a convex set equals ~ times the mean breadth 6 and prove an asymptotic formula for the expectation of the mean breadth of the above convex hull for N ~ oo. In analogy to [10] we show how this can be generalized for not necessarily uniformly distributed random points in K. By the duality principle of projective geometry we could apply a corresponding result in [10] to relate the expectation of the number of vertices of the convex hull of N random points in R 2 to the expectation of the number of edges ( = n u m b e r of vertices!) of a polygon generated by N random lines. But the following result cannot be applied in an analogous way, because the dual of the breadth of a convex set depends on the duality used. We call K a convex body in IRe, d > 2 , if K is a compact convex subset of ]Re with interior points. By b(u) we denote the breadth of K in the direction of the unit vector u, that is, the distance between the two supporting hyperplanes of K orthogonal to u, and we set b(K) for the mean value of b(.) with respect to the rotation invariant probabili ty measure co on the space U of all unit vectors of IRd. Let d(., .) denote the Euclidean distance in IR d. We remember that a real function g, defined on a set B in IRd, is C-Lipschitz-continuous on B, 0 < C < oo, if Ig(xl)--g(X2) [ ~ Cd(xl , X2) for all xi , x2~B.