EDIT: Oops. Followed a link and revived an old topic by mistake.

Perhaps this will help.

In three-space, the locus equidistant from...

... 1 point is a space.
... 2 coincident points is a space.
... 2 noncoincident points is a plane.
... 3 colinear points is the empty set.
... 3 noncolinear points is a line.
... 4 coplanar points is the empty set.
... 4 noncoplanar points is a point.
... >4 points (any coincident): evaluate as one of the other cases.
... >4 points (none coincident) is the empty set.