Vol. 60, No. 2, 1975

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Vol. 299: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
 
Other MSP Journals
A metric basis characterization of Euclidean space

Grattan Patrick Murphy

Vol. 60 (1975), No. 2, 159–163
Abstract

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
Milestones
Received: 2 July 1974
Published: 1 October 1975
Authors
Grattan Patrick Murphy