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

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 $L_p((x,y))=(\mathrm{\mid }x{\mathrm{\mid }}^p+\mathrm{\mid }y{\mathrm{\mid }}^p{)}^{1\mathrm{/}p}\mathrm{.}$ 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 $L_{\ast }((x,y))=\mathrm{\mid }xy\mathrm{\mid }$. 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”.