On farthest Bregman Voronoi cells

ABSTRACT

Let g be a strictly convex function on an evenly convex set $X\subset {\mathbb{R}}^n$ with nonempty interior. Assuming that g is differentiable on $intX,$ we consider the Bregman distance $D_g$ associated with g. Given a set T $\subset {\mathbb{R}}^n$, whose elements are called sites, and a particular site s, the farthest g-Bregman Voronoi cell of s, denoted by $F_T^g\left(s\right)$, consists of all points that are farther from s than from any other site with respect to $D_g$. In this paper we study farthest g-Bregman Voronoi cells; in particular, we characterize those sets that can be written as $F_T^g\left(s\right)$ for some $T\subset R^n$ and some $s\in T$.

Keywords:

• Voronoi cell
• Bregman distance
• farthest site

2010 Mathematics Subject Classifications:

• 52A20
• 26B25

Acknowledgments

Juan Enrique Martínez-Legaz gratefully acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities, through Grant PGC2018-097960-B-C21 and the Severo Ochoa Program for Centers of Excellence in R&D (CEX2019-000915-S). He is affiliated with MOVE (Markets, Organizations and Votes in Economics). We are indebted to an anonymous referee, whose careful reading of the manuscript and numerous helpful remarks has greatly helped us to correct and improve the presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Ministerio de Ciencia, Innovación y Universidades [CEX2019-000915-S, PGC2018-097960-B-C21].