### Abstract

© 2016 Elsevier Inc. All rights reserved. Given an arbitrary set T in the Euclidean space whose elements are called sites, and a particular site s, the Voronoi cell of s, denoted by VT(s), consists of all points closer to s than to any other site. The Voronoi mapping of s, denoted by ψs, associates to each set T ∋ s the Voronoi cell VT(s) of s w.r.t. T. These Voronoi cells are solution sets of linear inequality systems, so they are closed convex sets. In this paper we study the Voronoi inverse problem consisting in computing, for a given closed convex set F ∋ s, the family of sets T ∋ s such that ψs(T)=F. More in detail, the paper analyzes relationships between the elements of this family, ψs-1, and the linear representations of F, provides explicit formulas for maximal and minimal elements of ψs-1 (F), and studies the closure operator that assigns, to each closed set T containing s, the largest element of ψs-1 (F), where F = VT(s).

Original language | English |
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Pages (from-to) | 248-271 |

Journal | Linear Algebra and Its Applications |

Volume | 504 |

DOIs | |

Publication status | Published - 1 Sep 2016 |

### Keywords

- 15A39
- 51M20
- 52C22
- Inverse problem
- Linear inequality systems
- Voronoi cells
- Voronoi inverse mapping

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## Cite this

Goberna, M. A., Martínez-Legaz, J. E., & Vera De Serio, V. N. (2016). The Voronoi inverse mapping.

*Linear Algebra and Its Applications*,*504*, 248-271. https://doi.org/10.1016/j.laa.2016.04.004