The Limit of $$L_p$$ Voronoi Diagrams as $$p\rightarrow 0$$ is the Bounding-Box-Area Voronoi Diagram

Abstract

AbstractWe consider the Voronoi diagram of points in the real plane when the distance between two points a and b is given by $$L_p(a-b)$$ L p ( a - b ) where $$L_p((x,y)) = (|x|^p+|y|^p)^{1/p}.$$ 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 $$L_*((x,y)) = |xy|$$ L ∗ ( ( x , y ) ) = | x y | . 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 $$L_*$$ L ∗ as defined above the geometric $$L_0$$ L 0 distance.