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.

29 pages

References in corpus (1)

• Products of metric spaces, covering numbers, packing numbers and characterizations of ultrametric spaces