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

TL;DR

This paper studies the behavior of $L_p$ Voronoi diagrams as $p$ approaches zero, showing they converge to a new diagram based on a geometric $L_0$ distance involving hyperbolic bisectors.

Contribution

It establishes the limit of $L_p$ Voronoi diagrams as $p$ approaches zero, introducing the geometric $L_0$ distance and characterizing its bisectors.

Findings

Voronoi diagrams converge to a diagram with hyperbolic bisectors as $p o 0$

The limit diagram uses a new distance function $L_*$ involving $|xy|$

Bisectors in the limit diagram are composed of lines and hyperbolic branches

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 $L_*((x,y)) = |xy|$. 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".