Strict Convexity and Betweenness.

In this paper, we discuss the two concepts of betweeness in a metric linear space that arise from the vector space structure and from the metric space structure. We also explore the relation between the mid-points obtained from the algebraic structure and the metric structure in such spaces. We show that a real metric linear space is normable if and only if every algebraic mid-point is a metric mid-point if and only if algebraic betweeness implies metric betweeness. We also show that a real metric linear space is pseudo strictly convex if and only if the metric betweeness implies the algebraic betweeness. As a corollary, it turns out that a real metric linear space is normable with a strictly convex norm if and only if the notions of algebraic mid-points and metric mid-points coincide if and only if the notions of algebraic betweeness and metric betweeness coincide. [ABSTRACT FROM AUTHOR]