## Abstract

For a finite point set in R^{d}, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number L(X) of a given point set X is defined as the number of steps of the peeling process in order to delete all points in X. It is known that if X is a set of random points in R^{d}, then the expectation of L(X) is Θ(|X|^{2∕(d+1)}), and recently it was shown that if X is a point set of the square grid on the plane, then L(X)=Θ(|X|^{2∕3}). In this paper, we investigate the layer number of α-evenly distributed point sets for α>1; these point sets share the regularity aspect of random point sets but in a more general setting. The set of lattice points is also an α-evenly distributed point set for some α>1. We find an upper bound of O(|X|^{3∕4}) for the layer number of an α-evenly distributed point set X in a unit disk on the plane for some α>1, and provide an explicit construction that shows the growth rate of this upper bound cannot be improved. In addition, we give an upper bound of O(|X|[Formula presented]) for the layer number of an α-evenly distributed point set X in a unit ball in R^{d} for some α>1 and d≥3.

Original language | English |
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Article number | 112029 |

Journal | Discrete Mathematics |

Volume | 343 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2020 |

Externally published | Yes |

### Bibliographical note

Funding Information:Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07043049), and also by the Hankuk University of Foreign Studies, South Korea Research Fund.Partially supported by ISF, Israel Grant No. 2023464 and BSF, Israel Grant No. 2006099.

Publisher Copyright:

© 2020 Elsevier B.V.

## Keywords

- Convex layer
- Evenly-distributed point set
- Peeling sequence