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

Abstract

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 $L_{\ast }((x,y))=|xy|$. 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 $L_{\ast }$ as defined above the "geometric $L_0$ distance".

arXiv comment15 pages, 13 figures