Some remarks on M-convexity and best approximation

T.D. Narang

Abstract: To study the uniqueness of best approximation properties for $M$-convex subsets of metric spaces, strictly $M$-convex and uniformly $M$-convex metric spaces were introduced in [2] by using the notion of $M$-convexity in metric spaces. In this note it is shown that strictly $M$-convex and uniformly $M$-convex metric spaces do not serve any fruitful purpose for the uniqueness of solutions of best approximation problems (the very purpose for which these spaces were introduced) as these prove the uniqueness of best approximation problems only when they are Mengerian; however, Mengerian spaces in the sense of [2] do not exist. We also answer some of the problems raised in [2] and show that some of the results proved in [2] are incorrect.

Classification (MSC2000): 41A52; 41A50

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