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Local Euclidean four-point properties which characterize inner-product spaces


Authors: J. E. Valentine and S. G. Wayment
Journal: Proc. Amer. Math. Soc. 50 (1975), 337-343
MSC: Primary 52A50; Secondary 46C05
DOI: https://doi.org/10.1090/S0002-9939-1975-0370375-6
MathSciNet review: 0370375
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a complete, convex, externally convex metric space. We show $ M$ is an inner-product space if and only if for each point $ t$ of $ M,M$ contains a sphere $ {S_t}$ which has the euclidean queasy, feeble or weak four-point property.


References [Enhancements On Off] (What's this?)

  • [1] E. Z. Andalafte and L. M. Blumenthal, Metric characterizations of Banach and Euclidean spaces, Fund. Math. 55 (1964), 23-55. MR 29 #2625. MR 0165338 (29:2625)
  • [2] E. Z. Andalafte and J. E. Valentine, Intrinsic four point properties which characterize hyperbolic and euclidean spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 17-20. MR 0348636 (50:1133)
  • [3] L. M. Blumenthal, Theory and applications of distance geometry, Clarendon Press, Oxford, 1953. MR 14, 1009. MR 0054981 (14:1009a)
  • [4] -, An extension of a theorem of Jordan and von Neumann, Pacific J. Math. 5 (1955), 161-167. MR 16, 1139. MR 0070155 (16:1139d)
  • [5] H. Busemann, The geometry of geodesics, Academic Press, New York, 1955. MR 17, 779. MR 0075623 (17:779a)
  • [6] M. M. Day, On criteria of Kasahara and Blumenthal for inner-product spaces, Proc. Amer. Math. Soc. 10 (1959), 92-100. MR 21 #5140. MR 0106407 (21:5140)
  • [7] J. E. Valentine and S. G. Wayment, Wilson angles in linear normed spaces, Pacific J. Math. 36 (1971), 239-243. MR 43 #3775. MR 0278043 (43:3775)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0370375-6
Keywords: Banach space, complete, convex, externally convex, inner-product space, local euclidean four-point properties, metric space
Article copyright: © Copyright 1975 American Mathematical Society

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