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

Publication date: 15 July 2022

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)) = | 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_{*}

as defined above the "geometric

L_{0}

distance".

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