The limit of L_p Voronoi diagrams as p → 0 is the bounding-box-area Voronoi diagram

07/15/2022

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by   Herman Haverkort, et al.

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We consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by L_p(a-b) where L_p((x,y)) = (|x|^p+|y|^p)^1/p. We prove that the Voronoi diagram has a limit as p converges to zero from above or from below: it is the diagram that corresponds to the distance function L_*((x,y)) = |xy|. In this diagram, the bisector of two points in general position consists of a line and two branches of a hyperbola that split the plane into three faces per point. We propose to name L_* as defined above the "geometric L_0 distance".