On Metric Spaces in Which Metric Segments Have Unique Prolongations

An M-space is a metric space (X, d) having the property that for each pair of points p, q ∈ X with d(p, q) = λ and for each real number α ∈ [0, λ], there is a unique rα ∈ X such that d(p, rα) = α and d(rα, q) = λ − α. In an M-space (X, d), we say that metric segments have unique prolongations if points p, q, r, s satisfy d(p, q) + d(q, r) = d(p, r), d(p, q) + d(q, s) = d(p, s) and d(q, r) = d(q, s) then r = s. This paper mainly deals with some results on best approximation in metric spaces for which metric segments have unique prolongations. Rotundity or strict convexity has been studied extensively in Banach spaces (see e.g., [6]). It is well known that metric lines are unique in a Banach space B if and only if B is strictly convex ([1], [2], [3], [10]). This result is not valid in the metric space setting. There are complete convex, externally convex metric spaces (see [4], [5], [7]) in which the concepts of strict convexity and unique metric lines are not equivalent. However, Freese and Murphy [4], Freese, Murphy and Andalafte [5] and Khalil [7] have shown that unique metric lines (metric segments have unique prolongations) and strict convexity (redefined in purely metric terms) are equivalent in a larger class Received November 7, 2007 and in revised form Februray 5, 2008. AMS Subject Classification: Primary 41A65, Secondary 52A05, 51M30.