Strict convexity in M-spaces
A metric space (X,d) is called an M-space if for every \(p,q\in X\) with \(d(p,q)=\lambda\), and every \(\alpha\in [0,\lambda]\), there is a unique \(r_{\alpha}\in X\) such that \(d(p,r_{\alpha})=\alpha\) and \(d(r_{\alpha},q)=\lambda -\alpha\). The authors prove that the strict convexity of an M-space (X,d) is equivalent to the fact that every proximinal convex subset of X is a Chebyshev set and, also, to a monotonicity property of d. These characterizations remain valid in any complete, convex metric space. Finally, the concept of local M-space is introduced and, for this structure, local versions of the above results are proved.
scientific article; zbMATH DE number 4121001
scientific article