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. We prove that if XX is a metric space and A⊂XA\subset X satisfies the condition Nagata(n,c)\hspace{0.1em}\text{Nagata}\hspace{0.1em}\left(n,c), then any 1-Lipschitz map f:A→Yf:A\to Y to a Banach space YY admits a Lipschitz extension F:X→YF:X\to Y whose Lipschitz constant is at most 1,000⋅(c+1)⋅log2(n+2)\hspace{0.1em}\text{1,000}\hspace{0.1em}\cdot \left(c+1)\cdot {\log }_{2}\left(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⊂XA\subset X consists of nn points, then Lipschitz extensions as above exist with a Lipschitz constant of at most 600⋅logn⋅(loglogn)−1600\cdot \log n\cdot {\left(\log \log n)}^{-1}.

Keywords

• extension of lipschitz maps
• lipschitz n-connected
• nagata dimension
• spaces of non-positive curvature
• banach spaces
• primary 54c20
• secondary 54f45