Mutually nearest and farthest points of sets and the drop theorem in geodesic spaces
Let \((E,d)\) be a geodesic metric space [\textit{D. Burago, Yu. Burago} and \textit{S. Ivanov}, A course in metric geometry. Graduate Studies in Mathematics. 33. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0981.51016)] and \(A,\) \(X\in\mathcal{P}(E)\) two nonvoid subset of \(E.\) The authors study the problem \(\min(A,X)\) (respectively \(\max(A,X)\)) of finding \((a_{0} ,x_{0})\in A\times X\) such that \(d(a_{0},x_{0})=\inf\{d(a,x):\;a\in A,\) \(x\in X\}\) (respectively \(D(a_{0},x_{0})=\sup\{d(a,x):\;a\in A,\) \(x\in X\}.\) One obtains some generic results (with respect to the Hausdorff-Pompeiu metric) on the well-posedness of these problems in different geodesic metric spaces and under different conditions on the sets \(A\) (fixed) and \(X.\) For example: If \((E,d)\) is a complete Busemann convex geodesic metric space with curvature bounded below by \(k<0\) and with the geodesic extension property, then for \(A\in\mathcal{P}_{b,cl}(E),\) the set \(W_{\max}=\{X\in\mathcal{P} _{b,cl,cv}(E):\;\max A,X)\) is well posed\(\}\) is a dense \(G_{\delta}\)-set in \(\mathcal{P}_{b,cl,cv}(E)\) (Theorem 3.2.). Also, if \((E,d)\) is a complete geodesic metric space with no bifurcation geodesics and the geodesic extension property, the set \(W_{\max}=\{X\in\mathcal{P}_{cp}(E):\max(A,X)\) is well-posed is a dense \(G_{\delta}\)-set in \(\mathcal{P}_{cp}(E),\) for every \(A\) in \(\mathcal{P}_{cp}(E)\) (Theorem 4.2). Similar results are established for \(\min(A,X).\) Finally, a Drop Theorem in Busemann convex geodesic spaces is obtained (Th.5.3) and applied to an optimization result for convex functions.
scientific article; zbMATH DE number 6008825
scientific article