Special metrics (Q2782216)

This is a survey paper on recent results concerning several kinds of special metrics. There are three main topics: midset properties, topologically equivalent metrics with unusual properties, and ultrametrics. We briefly describe these topics. Given a metric space \((X, \rho)\), a set of the form \(M(y,z) =\{x\in X:\rho(x,y) = \rho(x,z)\}\) is called a midset, and a metric space is said to have the \(n\)-points midset property provided every midset has exactly \(n\) points (this topic is related to covering dimension and to coloring of simple graphs). The second topic springs from some well-known metrics with unusual properties introduced by \textit{J. Nagata} such as his result [Compos. Math. 15, 227-237 (1962; Zbl 0116.14402)] which yields that every metric space has a topologically equivalent metric for which the set of all \(\varepsilon\)-balls is closure preserving (for every \(\varepsilon>0)\) . The third topic, ultrametrics, arises in a large number of settings in mathematics. The discussion of ultrametrics in this paper concerns embeddings of ultrametric spaces into \(n\)-dimensional Euclidean spaces, embeddings into universal ultrametric spaces for a given weight, and a use of ultrametrics in domain theory. There are nine open questions stated in the paper.NEWLINENEWLINEFor the entire collection see [Zbl 0983.00046].