Vol. 60, No. 2, 1975

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A metric basis characterization of Euclidean space

Grattan Patrick Murphy

Vol. 60 (1975), No. 2, 159–163

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.

Mathematical Subject Classification
Primary: 52A50
Received: 2 July 1974
Published: 1 October 1975
Grattan Patrick Murphy