Minimal universal metric spaces

TL;DR

This paper investigates the conditions for the existence and uniqueness of minimal universal metric spaces and classes, providing constructions and characterizations for various classes including three-point and normed spaces.

Contribution

It introduces the concept of minimal universal metric spaces and classes, establishing conditions for their existence, uniqueness, and providing explicit examples.

Findings

Conditions for minimal $rak{M}$-universality are established.

Examples constructed for three-point and $n$-dimensional normed spaces.

Characterizations of minimal universal spaces for specific subclasses of metric spaces.

Abstract

Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions under which, for given metric space $X$, there is a class $\mathfrak{M}$ of metric spaces such that $X$ is minimal $\mathfrak{M}$-universal. We generalize the notion of minimal $\mathfrak{M}$-universal metric space to notion of minimal $\mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $\mathfrak{M}$ is minimal $\mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.