NEW METRIC CHARACTERIZATIONS OF BANACH SPACES

ABSTRACT. Recent characterizations of real strictly convex Banach spaces among complete, convex, externally convex metric spaces have made use of a metrization of the classical Theorem of Menelaus and its converse, and of a metrization of associativity of a natural addition operation defined in metric spaces with unique metric lines. In the present paper it is shown that a metrization of a weak form of associativity, (x + y) + (-y) = x, suffices to characterize strictly convex Banach spaces, and results are obtained which generalize the earlier characterizations. 1. In a recent characterization of real strictly convex Banach spaces, Freese and Murphy [3] show that the consistent midpoint property (CMP) suffices to characterize strictly convex Banach spaces among complete, convex, externally convex metric spaces with unique metric lines. A metric space M has the consistent midpoint property (CMP) provided for each p,q,r,s G M, m(m(p,q),m(r,s)) = m(m(p,r),m(q,s)), where m(x,y) denotes the metric midpoint of x and y. Independently Valentine [5] showed that a slight modification of CMP, the diagonal bisector property, provides a characterization of real strictly convex Banach spaces without the necessity of assuming uniqueness of metric lines. An almost immediate consequence of CMP is that if a zero element 0 6 M is fixed and an operation p + q is defined on M in the natural way, that operation is associative. In the present paper it is shown that weaker forms of CMP, one of wkich corresponds to the associativity property (x + y) + (-y) = x, suffice to characterize real strictly convex Banach spaces among complete, convex, externally convex metric spaces. Let M denote a complete, convex, externally convex metric space, let pq denote the distance between points p,q 6 M, and m(p,q) denote any metric midpoint of p an