Ultrametrics, extending of Lipschitz maps and nonexpansive selections (Q2862662)

A metric space \((X, d)\) is an ultrametric space if the metric is non-archimedean, or, an ultrametric, i.e., instead of the triangle inequality, \(d(x, z) \leq max(d(x, y), d(y,z))\) for all \(x, y, z \in\) X. Such spaces are important in the study of group actions on metric spaces and the classification of certain topological groups, as well as normed linear spaces over non-archimedean fields. For metric spaces \((X, d)\) and \((Y, \rho)\), a multifunction \(F : Y \rightarrow P_*(X)\), (\(P_*(x) = P(X) \setminus \{\emptyset\}\)), is \textit{shape Lipschitz} iff for all \(v, w \in Y\), there exists a Lipschitz constant \(M \geq 0\) such that \(d_H(F(v), F(w)) \leq M\rho(v,w)\), \(d_H\) is the Hausdorff distance induced by \(d\), and \(Lip(f)\) is the least Lipschitz constant. The paper shows that one of the characterizations of an ultrametric space is that \((X, d)\) is an ultrametric space iff for all \(c > 1\) and every closed set \(A \neq \emptyset\) of \(X\), there is a retraction \(r : X \rightarrow A\) (i.e., \(r(a) = a\), \(a \in A\)) such that \(Lip(r) \leq c\). Another characterization is that \((X,d)\) is ultrametric iff for every such multifunction \(F\) with \(d_H(F(v), F(w) )\leq \rho(v,w)\), for all \(v, w \in Y\), there is a function \(\phi : X \times Y \rightarrow X\) such that Lip(\(\phi\)) is arbitrarily close to 1 (\(X \times Y\) has the maximum metric induced by \(d\) and \(\rho\)) and for any \(x \in X\) and \(y \in Y\), \(\phi(x, \centerdot)\) is a selection for \(F\) (i.e., \(\phi(x, \centerdot) \in F(y)\)) and \(\phi (\centerdot, y)\) is a retraction of \(X\) onto \(F(y)\).NEWLINENEWLINEA closed set \(A \subset X\) is \textit{shape proximinal} in \(X\) if \(\forall x \in X\), \(\exists a \in A\) such that \(d(x, a) = dist_d(x,A)\); the space \((X,d)\) has the \textit{shape absolute proximinality property} (APP) if every nonempty closed set in \(X\) is proximinal. The paper shows that \((X,d)\) is an ultrametric space with APP iff every nonempty closed set \(A\) of \(X\) is a retract of \(X\) under a \textit{shape nonexpansive} map \(r\) (i.e., \(Lip(r) \leq 1\)).NEWLINENEWLINEAn ultrametric space \((X,d)\) is \textit{shape standard} iff \(d(X \times X) \setminus \{0\}\) is discrete in \((0, +\infty)\). Every compact ultrametric space \((X,d)\) is standard and every order on \(X\) is \textit{shape compatible with the metric}, i.e., any total order \(``\leq''\) on \(X\) such that \(\forall x, y, z \in X\), \(x \leq y \leq z\) iff \(d(x,z) = max(d(x,y), d(y,z))\). Every compact ultrametric space is standard and every order on the space compatible with the metric makes the space \textit{shape topologically well ordered}, i.e., every closed nonempty subset has a least element with respect to the order.NEWLINENEWLINEThe paper also deals with some partial orders in the set \(\mathcal{K}\) of all compact metric spaces up to isometry. The set \(\mathcal{K}_* = \mathcal{K} \setminus \{\emptyset\}\) with the \textit{shape Gromov-Hausdorff} metric is a separable complete metric space. For \(K, L \in \mathcal{K}_*\), \(d_{GH}(K, L)\) is the infimum over all Hausdorff distances of isometric copies of \textit{shape K} and \textit{shape L} in any metric space. The paper considers three classical partial orders on the set \(\mathcal{K}\). Using the notion of \textit{shape uniformly compact}, Gromov had shown that such subfamilies are upper bounds with respect to one of the three partial orders. This paper shows that the same condition is equivalent to the upper boundedness with respect to the other two orders.