Plastic pairs of metric spaces

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 $\tilde{x},\tilde{y}\in X$ for which $d(f(\tilde{x}),f(\tilde{y}))<d(\tilde{x},\tilde{y})$. We give sufficient conditions for this property and for its uniform version in terms of finite $\epsilon$-nets and finite $\epsilon$-separated subsets.