The Ceva property characterizes real, strictly convex Banach spaces

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.