On approximative properties of sets in special metric spaces
Let \((X,d)\) be a convex metric space, i.e., each pair \(x,y\in X\) is connected by a unique geodesic isometric to the interval \([0,d(x,y)] \subset \mathbb{R}\). Then for every \(\lambda\in \mathbb{R}\) the function \(w_\lambda: X\times X\to \mathbb{R}_+\) is well defined by the conditions: (a) \(d(x,w_\lambda (x,y))=|\lambda|d(x,y)\); (b) \(w_\lambda(x,y)\) belongs to a straight line (unbounded geodesic) determined by \(x,y\). Assume that for each \(\lambda\) and a closed ball \(B\) \(w_\lambda\) is uniformly continuous on \(B\times B\). Assume also that each ball \(B_R(x)\) is strictly convex in a sense that it is convex and for each \((x_n)\), \((y_n)\) in \(B_R(x)\) \(d(x_n,y_n)\to 0\), \(n\to\infty\) provided \(d(x, w_{1/2} (x_n,y_n))\to 0\), \(n\to\infty\). Theorem 1. Every closed convex set of \(X\) is an approximately compact Chebyshev set. Theorem 2. Let \(P_M(x): =\{y\in M;d (x,y)= d(x,M)\}\) be the metric projection on \(M\neq\emptyset\). Then the set \(\{x\in X; \#P_M(x)=1\}\) is residual.
scientific article; zbMATH DE number 1541490