We consider the Voronoi diagram of points in the real plane when the distance
between two points a and b is given by Lpβ(aβb) where Lpβ((x,y))=(β£xβ£p+β£yβ£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 Lββ((x,y))=β£xyβ£. 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
Lββ as defined above the "geometric L0β distance".Comment: 15 pages, 13 figure