Metric products and continuation of isotone functions

Abstract

Let $\mathbb{R}_+=[0,\infty)$ and let $A\subseteq\mathbb{R}^n_+$. We have found the necessary and sufficient conditions under which a function $\Phi:A\to\mathbb{R}_+$ has an isotone subadditive continuation on $\mathbb{R}^n_+$. It allows us to describe the metrics, defined on the Cartesian product $X_1\times...\times X_n$ of given metric spaces $(X_1,d_{X_1}),...,(X_n,...,d_{X_n})$, generated by the isotone metric preserving functions on $\mathbb{R}^n_+$. It also shows that the isotone metric preserving functions $\Phi:\mathbb{R}^n_+\to\mathbb{R}_+$ coincide with the first moduli of continuity of the nonconstant bornologous functions $g:\mathbb{R}^n_+\to\mathbb{R}_+$. We discuss some algebraic properties of sets $X\subseteq \mathbb{R}$ providing the existence of isometric embeddings $f:B\to X$ for every three-point $B\subseteq \mathbb{R}$. In particular, we prove that every finite subset of $\mathbb{R}$ is isometric to some subset of transcendental real numbers.