  The midpoint set M(S) of a set S of points is the set of all midpoints of
pairs of points in S. We study the largest cardinality of a midpoint set M(S)
in a finite-dimensional normed space, such that M(S) is contained in the unit
sphere, and S is outside the closed unit ball. We show in three dimensions that
this maximum (if it exists) is determined by the facial structure of the unit
ball. In higher dimensions no such relationship exists. We also determine the
maximum for euclidean and sup norm spaces.
