Absolute Lipschitz extendability and linear projection constants

Abstract

In this mostly expository article, we give streamlined proofs of several well-known Lipschitz extension theorems. We pay special attention to obtaining statements with explicit expressions for the Lipschitz extension constants. One of our main results is an explicit version of a very general Lipschitz extension theorem of Lang and Schlichenmaier. A special case of the theorem reads as follows. We prove that if X is a metric space and A ⊂ X satisfies the condition Nagata(n, c), then any 1-Lipschitz map f : A → Y to a Banach space Y admits a Lipschitz extension F : X → Y whose Lipschitz constant is at most 1000 • (c + 1) • log 2 (n + 2). By specifying to doubling metric spaces, this recovers an extension result of Lee and Naor. We also revisit another theorem of Lee and Naor by showing that if A ⊂ X consists of n points, then Lipschitz extensions as above exist with a Lipschitz constant of at most 600 • log n • (log log n) −1 .