A Universal Inequality for Stability of Coarse Lipschitz Embeddings

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 * ∈ Lip 0 ( X ) there exist α, γ > 0 depending only on f and Q x * ∈ Lip 0 ( Y ) with ‖ Q x * ‖ L i p ≤ α ‖ x * ‖ L i p such that | Q x * f ( x ) − x * ( x ) | ≤ γ ‖ x * ‖ L i p , f o r a l l x ∈ 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.