On functions preserving regular semimetrics and quasimetrics satisfying the relaxed polygonal inequality (Q783249)

Let \(X\) be a non-empty set. A function \(d:X\times X\to\mathbb{R}_+\) is called a \textit{semi-metric} if it satisfies the following two conditions: \begin{itemize} \item[(S1)] \(d(x,y)=0\) iff \(x=y\); \item[(S2)] \(d(x,y)=d(y,x)\) for all \(x,y\in X\). \end{itemize} In the paper under review, the authors study non-negative functions \(f:\mathbb{R}_+\to\mathbb{R}_+\) which preserve some types of semi-metrics, like: \textit{metric}, \textit{ultra-metric}, \textit{quasi-metric}, \textit{semi-metrics satisfying the relaxed polygonal inequality}, or \textit{semi-metrics satisfying the Chittenden condition}. The authors obtain characterizations of new classes of such functions. \begin{itemize} \item First, they give a precise characterization of functions preserving the regularity of a semimetric space in the sense of \textit{M. Bessenyei} and \textit{Z. Páles} [J. Nonlinear Convex Anal. 18, No. 3, 515--524 (2017; Zbl 1396.54037)]. \item Next, they show one of the main results of the article: for a non-decreasing function \(f:\mathbb{R}_+\to\mathbb{R}_+\) the following statements are equivalent: \begin{itemize} \item[(i)] for any semi-metric space \((X, d)\), if \(d\) satisfies the relaxed polygonal inequality, then so does \(f \circ d\); \item[(ii)] there exist a constant \(c\ge 1\) and a subadditive function \(g : [0,+\infty) \to [0,+\infty)\) such that \(g^{-1}(\{0\}) = \{0\}\) and \(g \le f \le cg\). \end{itemize} \item Finally, the authors revisit the theorem of \textit{P. Pongsriiam} and \textit{I. Termwuttipong} [Abstr. Appl. Anal. 2014, Article ID 163258, 9 p. (2014; Zbl 1440.54010)], in order to provide a simpler proof of their result characterizing functions which transform metric spaces into ultrametric ones. \end{itemize}