Extending Lipschitz functions via random metric partitions
The method of stochastic metric decomposition is applied to a number of problems concerning the extension of Lipschitz functions. For example, it is shown that there is a universal constant \(C>0\) such that for every metric space \(X\) its absolute extendability constant \(ae(X)\) does not exceed \(C \log \lambda(X)\), where \(\lambda(X)\) (the doubling constant of \(X\)) is the infimum of those \(\lambda\) so that every ball in \(X\) can be covered by \(\lambda\) balls of half the radius. For \(n\)-point metric spaces, the asymptotic growth of \(ae(X)\) can be estimated from above by \(C\log n / \log \log n .\) (\(ae(X)\) is the infimum of those \(K>0\) such that for every metric space \(Y \supset X\) and for every metric space \(Z\), every Lipschitz function \(f : X \to Z\) can be extended to the whole of \(Y\) with a no more than \(K\) times bigger Lipschitz constant.)
scientific article; zbMATH DE number 2156039
scientific article