Given a set S of n points in R d, 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 E d (k,n), the expected number of k-simplices when S is a random sample of n points from a probability distribution P on R d . When P is spherically symmetric we prove that E d (k, n)≤cn d-1 When P is uniform on a convex body K⊂R 2 we prove that E 2 (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 R 2 for which E 2((n-2)/2, n) is cn log n.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics