A Universal Inequality for Stability of Coarse Lipschitz Embeddings

Abstract

Let X and Y be two pointed metric spaces. In this article, we give a generalization of the Cheng–Dong–Zhang theorem for coarse Lipschitz embeddings as follows: If f:X → Y is a standard coarse Lipschitz embedding, then for each x* ∈ Lip₀(X) there exist α, γ > 0 depending only on f and Q_x* ∈ Lip₀(Y) with \({\Vert{{Q_{{x^*}}}}\Vert_{{\rm{Lip}}}} \le \alpha {\Vert {{x^*}}\Vert_{{\rm{Lip}}}}\) such that

$$\Vert{{Q_{{x^*}}}f(x) - {x^*}(x)}\Vert\le \gamma {\left\| {{x^*}} \right\|_{{\rm{Lip}}}},\;\;\;\;\;{\rm{for}}\;{\rm{all}}\;x \in X.$$

Coarse stability for a pair of metric spaces is studied. This can be considered as a coarse version of Qian Problem. As an application, we give candidate negative answers to a 58-year old problem by Lindenstrauss asking whether every Banach space is a Lipschitz retract of its bidual. Indeed, we show that X is not a Lipschitz retract of its bidual if X is a universally left-coarsely stable space but not an absolute cardinality-Lipschitz retract. If there exists a universally right-coarsely stable Banach space with the RNP but not isomorphic to any Hilbert space, then the problem also has a negative answer for a separable space.