The Ceva property characterizes real, strictly convex Banach spaces

Authors:
J. E. Valentine and S. G. Wayment

Journal:
Proc. Amer. Math. Soc. **78** (1980), 559-567

MSC:
Primary 51F99; Secondary 46B20, 52A01

DOI:
https://doi.org/10.1090/S0002-9939-1980-0556633-2

MathSciNet review:
556633

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Abstract | References | Similar Articles | Additional Information

Abstract: If *a, b, c* are distinct collinear points of a metric space, then

*p, q, r*, if

*s,t,u*are points distinct from

*p, q, r*on the metric lines , and , respectively, then the metric lines , and have a common point if and only if , and . In the euclidean plane, the requirement that forces the lines and to have a common point. Thus the case of parallel lines is avoided and the Ceva Property is meaningful in an arbitrary metric space. The main result of the paper is that a complete, convex, externally convex, metric space is a strictly convex Banach space over the reals if and only if it satisfies the Ceva Property.

**[1]**E. Z. Andalafte and L. M. Blumenthal,*Metric characterizations of Banach and Euclidean spaces*, Fund. Math.**55**(1964), 23-55. MR**0165338 (29:2625)****[2]**L. M. Blumenthal,*Theory and applications of distance geometry*, Clarendon Press, Oxford, 1953. MR**0054981 (14:1009a)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0556633-2

Keywords:
Banach space,
Ceva Property,
convex,
externally convex,
metric space

Article copyright:
© Copyright 1980
American Mathematical Society