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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Criteria for unique metric lines in Banach spaces


Authors: E. Z. Andalafte and J. E. Valentine
Journal: Proc. Amer. Math. Soc. 39 (1973), 367-370
MSC: Primary 52A50; Secondary 46B05
MathSciNet review: 0313947
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Abstract: A metric space $ M$ has the monotone property if for each point $ p$ and line $ L$ of $ M$ the distance $ px$ between $ p$ and a point $ x$ of $ L$ is monotone increasing as $ x$ recedes along either half-line of $ L$ determined by the foot of $ p$ on $ L$. It is shown that a Banach space (over the reals) has the monotone property if and only if it has unique metric lines. Using previously known results, additional equivalents of the monotone property are obtained and new proofs of some older criteria for unique metric lines result.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0313947-5
Keywords: Banach space, Chebyshev sets, metric lines, monotone property, rotundity, Young property
Article copyright: © Copyright 1973 American Mathematical Society