Metric products and continuation of isotone functions

TL;DR

This paper characterizes when functions on positive reals can be extended to isotone subadditive functions on the entire positive orthant, with applications to metrics and isometric embeddings involving transcendental numbers.

Contribution

It provides necessary and sufficient conditions for extending functions to isotone subadditive functions, linking metric-preserving functions to moduli of continuity, and explores algebraic properties of subsets of real numbers.

Findings

Characterization of functions with isotone subadditive extensions.

Description of metrics generated by isotone metric preserving functions.

Existence of isometric embeddings of finite subsets into transcendental numbers.

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.