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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Ceva property characterizes real, strictly convex Banach spaces
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by J. E. Valentine and S. G. Wayment PDF
Proc. Amer. Math. Soc. 78 (1980), 559-567 Request permission

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.
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Additional Information
  • © Copyright 1980 American Mathematical Society
  • 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