On the expected number of k-sets

Abstract

Given a set S of n points in Rd, a subset X of size d is called a k-simplex if the hyperplane aff(X) has exactly k points on one side. We study Ed (k,n), the expected number of k-simplices when S is a random sample of n points from a probability distribution P on Rd . When P is spherically symmetric we prove that Ed (k, n)≤cnd-1 When P is uniform on a convex body K⊂R2 we prove that E2 (k, n) is asymptotically linear in the range cn≤k≤n/2 and when k is constant it is asymptotically the expected number of vertices on the convex hull of S. Finally, we construct a distribution P on R2 for which E2((n-2)/2, n) is cn log n. © 1994 Springer-Verlag New York Inc.