  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".
