On Lipschitz partitions of unity and the Assouad–Nagata dimension
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Lipschitz extension theorems with explicit constants
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 X X is a metric space and A ⊂ X A\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 → Y f:A\to Y to a Banach space Y Y admits a Lipschitz extension F : X → Y F:X\to Y whose Lipschitz constant is at most 1,000 ⋅ ( c + 1 ) ⋅ log 2 ( 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 ⊂ X A\subset X consists of n n points, then Lipschitz extensions as above exist with a Lipschitz constant of at most 600 ⋅ log n ⋅ ( log log n ) − 1 600\cdot \log n\cdot {\left(\log \log n)}^{-1} .
Lipschitz extension theorems with explicit constants
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 X X is a metric space and A ⊂ X A\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 → Y f:A\to Y to a Banach space Y Y admits a Lipschitz extension F : X → Y F:X\to Y whose Lipschitz constant is at most 1,000 ⋅ ( c + 1 ) ⋅ log 2 ( 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 ⊂ X A\subset X consists of n n points, then Lipschitz extensions as above exist with a Lipschitz constant of at most 600 ⋅ log n ⋅ ( log log n ) − 1 600\cdot \log n\cdot {\left(\log \log n)}^{-1} .
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