## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 243-263 |

Number of pages | 21 |

Journal | Discrete & Computational Geometry |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1994 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics