Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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The Ceva property characterizes real, strictly convex Banach spacesHTML articles powered by AMS MathViewer

by J. E. Valentine and S. G. Wayment
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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