## Abstract

For (Formula presented.), let S be a set of points in (Formula presented.) in general position. A set I of k points from S is a k-island in S if the convex hull (Formula presented.) of I satisfies (Formula presented.). A k-island in S in convex position is a k-hole in S. For (Formula presented.) and a convex body (Formula presented.) of volume 1, let S be a set of n points chosen uniformly and independently at random from K. We show that the expected number of k-holes in S is in (Formula presented.). Our estimate improves and generalizes all previous bounds. In particular, we estimate the expected number of empty simplices in S by (Formula presented.). This is tight in the plane up to a lower-order term. Our method gives an asymptotically tight upper bound (Formula presented.) even in the much more general setting, where we estimate the expected number of k-islands in S.

Original language | English |
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Pages (from-to) | 308-326 |

Number of pages | 19 |

Journal | Random Structures and Algorithms |

Volume | 60 |

Issue number | 3 |

DOIs | |

State | Published - 1 May 2022 |

Externally published | Yes |

## Keywords

- convex position
- k-hole
- k-island
- random point set
- stochastic geometry

## ASJC Scopus subject areas

- Software
- General Mathematics
- Computer Graphics and Computer-Aided Design
- Applied Mathematics