On Farthest Bregman Voronoi Cells

J. E. Martínez Legaz, E. Naraghirad, M. Tamadoni Jahromi

Given the Bregman distance induced by a differentiable strictly convex function and a set of points, called sites, the farthest Bregman Voronoi cell of a particular site is the set of points for which the farthest site with respect to the considered Bregman distance is the given site. We obtain new expressions for farthest Bregman Voronoi cells, as well as conditions for their nonemptiness. We also give conditions for a closed convex set to be a cell. Moreover, using a minimax theorem due to B. Ricceri, we extend his result on the existence of two different intersecting cells, using essentially the same approach.

Keywords: Voronoi cell, Bregman distance, farthest site

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