The Transitive Property of Parallel Lines is a Characteristic Property of Real Strictly Convex Banach Spaces

J. E. Valentine
1985 Proceedings of the American Mathematical Society  
In a recent paper Freese and Murphy said a complete, convex, externally convex metric space has the vertical angle property provided for each four of its distinct points p, q, r, s, if m is a midpoint of p and q and of r and s. then pr = qs. In this paper we say a line L is parallel to a line N in such a space provided L and N contain points p, r, and q, s, respectively, such that the segments S(p,q) and S(r, s) have a common midpoint m. We further assume that if line L is parallel to line ¿V
more » ... rallel to line ¿V and line ¿V is parallel to line R, then L is parallel to R. The main result of this paper is that such a space is a real strictly convex Banach space. Since real strictly convex Banach spaces have all of the above properties, the characterization is then complete.
doi:10.2307/2045851 fatcat:xuixg6wjcrayhaydvuudoe7r2m