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We consider the Voronoi diagram of points in the real plane when the distancebetween 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 correspondsto the distance function $L_*((x,y)) = |xy|$. In this diagram, the bisector oftwo points in general position consists of a line and two branches of ahyperbola that split the plane into three faces per point. We propose to name$L_*$ as defined above the "geometric $L_0$ distance".

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