Consider any collection of non-overlapping discs of equal size (possibly just isolated points). We associate to this collection a partition of the plane into regions called The partition we associate to any distribution of non-overlapping discs in the plane is its Voronoi partition. We shall show that |
This figure is live: the discs are movable. |

There are a few characteristic properties of Voronoi cells that we shall require.

(1) Since all the discs are of equal size, a disc is contained within its Voronoi cell. |

| (2) If |

(3) For a layout of three discs, the Voronoi cells will be simplicial cones with a common vertex at one point - the | | |

| Suppose we fix two of the three discs and ask how this triple point varies as the third disc moves around. If the two discs are far enough apart, the triple point can be anywhere on the line bisecting the given pair. But as soon as they are close enough together, there will be a The figure also illustrates that the caps drawn from the triple point don't overlap (except possibly along their edges), and that their common vertex angle can be at most one-third of 120^{o} |