The *dispersion*^{5.6}of a finite set of samples in a metric space is^{5.7}

Figure 5.4 gives an interpretation of the definition for
two different metrics. An alternative way to consider dispersion is
as the radius of the largest empty ball (for the metric,
the balls are actually cubes). Note that at the boundary of (if
it exists), the empty ball becomes truncated because it cannot exceed
the boundary. There is also a nice interpretation in terms of Voronoi
diagrams. Figure 5.3 can be used to help explain
dispersion in
. The *Voronoi vertices* are the points at which three or more Voronoi regions meet.
These are points in for which the nearest sample is far. An
open, empty disc can be placed at any Voronoi vertex, with a radius
equal to the distance to the three (or more) closest samples. The
radius of the largest disc among those placed at all Voronoi vertices
is the dispersion. This interpretation also extends nicely to higher
dimensions.

Steven M LaValle 2020-08-14