### 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) |
---|---|

Pages (from-to) | 243-263 |

Number of pages | 21 |

Journal | Discrete & Computational Geometry |

Volume | 11 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 1994 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*Discrete & Computational Geometry*,

*11*(1), 243-263. https://doi.org/10.1007/BF02574008

}

*Discrete & Computational Geometry*, vol. 11, no. 1, pp. 243-263. https://doi.org/10.1007/BF02574008

**On the expected number of k-sets.** / Bárány, Imre; Steiger, William.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the expected number of k-sets

AU - Bárány, Imre

AU - Steiger, William

PY - 1994/12/1

Y1 - 1994/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=51249163425&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249163425&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/BF02574008

DO - https://doi.org/10.1007/BF02574008

M3 - Article

VL - 11

SP - 243

EP - 263

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -