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: If *a, b, c* are distinct collinear points of a metric space, then \[ (a,b,c) = \left \{ {\begin {array}{*{20}{c}} {ab/bc} \hfill & {{\text {if}}\;b\;{\text {is}}\;{\text {between}}\;a\;{\text {and}}\;c,} \hfill \\ { - (ab/bc)} \hfill & {{\text {otherwise}}.} \hfill \\ \end {array} } \right .\] A metric space satisfies the Ceva Property provided for each triple of noncollinear points *p, q, r*, if *s,t,u* are points distinct from *p, q, r* on the metric lines $L(p,q),L(q,r)$, and $L(r,p)$, respectively, then the metric lines $L(p,t),L(q,u)$, and $L(r,s)$ have a common point if and only if $(p,s,q)(q,t,r)(r,u,p) = 1$, and $pq/ps \ne pu/pr$. In the euclidean plane, the requirement that $pq/ps \ne pu/pr$ forces the lines $L(r,s)$ and $L(q,u)$ 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.

- E. Z. Andalafte and L. M. Blumenthal,
*Metric characterizations of Banach and Euclidean spaces*, Fund. Math.**55**(1964), 23–55. MR**165338**, DOI https://doi.org/10.4064/fm-55-1-23-55 - Leonard M. Blumenthal,
*Theory and applications of distance geometry*, Oxford, at the Clarendon Press, 1953. MR**0054981**

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

Article copyright:
© Copyright 1980
American Mathematical Society