Optimal quasi-metrics in a given pointwise equivalence class do not always exist (Q2515550)

Throughout this paper a quasi-metric on a set \(X\) will be any function \(\rho:X\times X\to [0,\infty]\) satisfying: {\parindent=0.6cm\begin{itemize}\item[--] \(\rho\) is non-degenerate, i.e., \(\rho(x,y)=0\) iff \(x=y\). \item[--] \(\rho\) is quasi-symmetric, i.e., \(\widetilde{C}_{\rho}:= \sup\{\frac{\rho(x,y)}{\rho(y,x)}:x, y \in X, x\neq y\}<\infty\). \item[--] \(\rho\) is quasi-ultrametric, i.e., \(C_{\rho}:= \sup\{\frac{\rho(x,y)}{\max\{\rho(x,z),\rho(z,y)\}} :x, y,z \in X, \text{not all equal}\}<\infty\). \end{itemize}} It is easy to see that \(\widetilde{C}_{\rho}, C_{\rho}\in [1, \infty)\), and when \(\rho\) is in particular a metric on \(X\), then \(\widetilde{C}_{\rho}=1\) and \(C_{\rho}\in [1,2]\). Moreover, if \(\widetilde{C}_{\rho}=1\) and \(C_{\rho}=1\), it follows that \(\rho\) is in fact an ultrametric. Analogously to metrics, any quasi-metric \(\rho\) induces in a natural way a topology \(\tau_\rho\) on \(X\). But contrary to the metric setting, it is known that quasi-metrics possibly fail to be continuous and their associated balls to be open in their induced topology. On the other hand, two quasi-metrics \(\rho\) and \(\varrho\) on \(X\) are called pointwise equivalent (\(\rho\approx\varrho\)) if there exists a constant \(C\in [1, \infty)\) such that \(C^{-1}\rho(x,y)\leq \varrho(x,y)\leq C \rho(x,y)\), for every \(x,y\in X\). In this scenario, the authors investigate some kind of optimization problems, for instance, given a quasi-metric \(\rho\) whether there exists or not another pointwise equivalent \(\varrho\) which is most ultrametric-like. In other words, is the infimum \(\inf\{C_\varrho: \varrho \approx \rho\}\) actually attained? This question was posed in [\textit{D. Mitrea} et al., Groupoid metrization theory. With applications to analysis on quasi-metric spaces and functional analysis. New York, NY: Birkhäuser/Springer (2013; Zbl 1269.46002)]. In this respect, the main result (Theorem 3.1) says that, in general, such a minimizer does not exist. In order to establish this, it is necessary to carry out a series of steps, including the development of the notion of Rolewicz-Orlicz spaces (which are topological vector spaces). Finally it is proved that with some additional properties the above optimization problem has a global minimizer (Theorem 7.4).