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Minimal arcs in metric spaces

Published online by Cambridge University Press:  09 April 2009

S. K. Hildebrand  and
Harold Willis Milnes
S. K. Hildebrand
Affiliation:
Texas Tech UniversityLubbock, Texas 79409, U.S.A.
Harold Willis Milnes
Affiliation:
Texas Tech UniversityLubbock, Texas 79409, U.S.A.
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In this paper we discuss the existence of an are of minimal length joining two arbitrary, yet fixed points in a complete metric space, where the metric is restricted only by the properties (A) and (B) given below. It is shown that under these conditions an arc of least length joining any two fixed points exists, and is unique. In addition, its length is shown to be equal to the metrie distance between the points.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 4 , June 1975 , pp. 426 - 430
Copyright
Copyright © Australian Mathematical Society 1975