Lipschitz extension theorems with explicit 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 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: If $X$ is any metric space and $A\subset X$ satisfies the condition $\text{Nagata}(n,c)$, then any $1$-Lipschitz map $f:A\to Y$ to a Banach space $Y$ admits a Lipschitz extension $F:X\to Y$ whose Lipschitz constant is at most $1000\cdot (c+1)\cdot {\log }_2(n+2)$. 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 $A\subset X$ consists of $n$ points, then Lipschitz extensions as above exist with a Lipschitz constant of at most $600\cdot \log n\cdot (\log \log n{)}^{-1}$.

Final version, to appear in Analysis and Geometry in Metric Spaces (AGMS)