2012/03/01 by Oleksiy Dovgoshey, O. Dovgoshey, Evgeniy Petrov +5
Mathematics · #26B25 #Advanced Topology and Set Theory #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Primary 54E35 #Secondary 26A15 #math.MG #msc:26A15 #msc:26B25 #msc:54E35
paper · pdf · doi:10.48550/arxiv.1203.0257
29 pages
arxiv created 2012/03/01 · openalex publication_date 2012/03/01 · arxiv updated 2012/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ℝ+=[0,∞) and let A⊆ℝn+. We have found the necessary and sufficient conditions under which a function Φ:A→ℝ+ has an isotone subadditive continuation on ℝn+. It allows us to describe the metrics, defined on the Cartesian product X1×...× Xn of given metric spaces (X1,dX1),...,(Xn,...,dXn), generated by the isotone metric preserving functions on ℝn+. It also shows that the isotone metric preserving functions Φ:ℝn+→ℝ+ coincide with the first moduli of continuity of the nonconstant bornologous functions g:ℝn+→ℝ+. We discuss some algebraic properties of sets X⊆ ℝ providing the existence of isometric embeddings f:B→ X for every three-point B⊆ ℝ. In particular, we prove that every finite subset of ℝ is isometric to some subset of transcendental real numbers.