Criteria for unique metric lines in Banach spaces
HTML articles powered by AMS MathViewer
- by E. Z. Andalafte and J. E. Valentine PDF
- Proc. Amer. Math. Soc. 39 (1973), 367-370 Request permission
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.References
- E. Z. Andalafte and L. M. Blumenthal, Metric characterizations of Banach and Euclidean spaces, Fund. Math. 55 (1964), 23–55. MR 165338, DOI 10.4064/fm-55-1-23-55
- Leonard M. Blumenthal, Theory and applications of distance geometry, Oxford, at the Clarendon Press, 1953. MR 0054981
- D. F. Cudia, Rotundity, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 73–97. MR 0155166
- Raymond W. Freese, Criteria for inner product spaces, Proc. Amer. Math. Soc. 19 (1968), 953–958. MR 227868, DOI 10.1090/S0002-9939-1968-0227868-8
- W. H. Young, On the Analytical Basis of Non-Euclidian Geometry, Amer. J. Math. 33 (1911), no. 1-4, 249–286. MR 1506126, DOI 10.2307/2369994
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 367-370
- MSC: Primary 52A50; Secondary 46B05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0313947-5
- MathSciNet review: 0313947