Publications (1)0 Total impact
-
[show abstract]
[hide abstract]
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.
03/2012;
