In En there is exactly one line
containing a given pair of points and exactly one k-flat containing a given
k-simplex (k + 1 points not contained in a lower dimensional space). The
purpose of this paper is to prove converses of these propositions in the setting
of complete, convex metric spaces. The most striking of these is given in
Theorem 1 where it is proved that a complete, convex metric space which
can be uniquely determined by any pair of points must be isometric with
a subset of the real line. Theorem 2 is a higher dimensional analogue of
this theorem. Metric characterizations of E1 and En are derived from these
results.
