Plastic pairs of metric spaces

TL;DR

This paper investigates pairs of metric spaces with a property that relates the existence of points with increased or decreased distances under any mapping, providing conditions for this property using finite nets and separated sets.

Contribution

It introduces a new property of metric space pairs, called plasticity, and establishes sufficient conditions for this property and its uniform version.

Findings

Characterization of plastic pairs via finite $ ext{epsilon}$-nets.

Conditions for uniform plasticity based on finite separated sets.

Theoretical framework for understanding metric space mappings.

Abstract

We address pairs $(X, Y)$ of metric spaces with the following property: for every mapping $f: X \to Y$ the existence of points $x, y \in X$ with $d(f(x),f(y)) > d(x,y)$ implies the existence of $\widetilde{x}, \widetilde{y}\in X$ for which $d(f(\widetilde{x}),f(\widetilde{y})) < d(\widetilde{x},\widetilde{y})$. We give sufficient conditions for this property and for its uniform version in terms of finite $\varepsilon$-nets and finite $\varepsilon$-separated subsets.