Extensions of Lipschitz mappings into a Hilbert space. (English) Zbl 0539.46017

Contemp. Math. 26, 189-206 (1984).
The main result of this paper is the following theorem: Let \(X\supset Y\) be metric spaces with \(| Y| =cardinality\) of \(Y=n\). Then every map f from Y into the Hilbert space \(\ell_ 2\) can be extended to a Lipschitz map \(\hat f\) from X into \(\ell_ 2\) with \(\| \hat f\leq c\sqrt{\log n}\| f\|\) where c is an absolute constant and \(\|\|\) denotes the Lipschitz norm of the map (i.e. \(\| f\| =\sup \{\| f(u)-f(v)\| /d(u,v);\) \(u,v\in Y)\). An example is presented in which it is shown that there exist for every n, metric spaces \(X\supset Y\) with \(| Y| =n\) and maps \(f: Y\to \ell_ 2\) so that for every extension \(\hat f\) of f we have \(\| \hat f\| \geq d\sqrt{\log n} \| f\| /\sqrt{\log \log n}\) for some absolute constant d. The main step in the proof of the theorem is the following geometric lemma.
For every \(\delta>0\) there is a \(c(\delta)\) so that if \(\{x_ i\}^ n_{i=1}\) are n points in \(\ell_ 2\) there are points \(\{u_ i\}^ n_{i=1}\) in the k-dimensional Hilbert space \(\ell^ k_ 2\) where \(k=[c(\delta)\quad \log n]\) so that for all \(1\leq i,j\leq n\) \[ \| x_ i-x_ j\| \leq \| u_ i-u_ j\| \leq(1+\delta)\| x_ i-x_ j\|. \] In a subsequent paper by the authors jointly with G. Schechtman it is proved that if the range space of f is a general Banach space (instead of \(\ell_ 2)\) an extension \(\hat f\) can be chosen so that \[ \| \hat f\| \leq c_ 1 \log n \| f\|. \] Problems 3 and 4 at the end of the paper were solved by J. Bourgain who proved that if Y is a metric space with \(| Y| =n\) then there is a subset \(Z\subset \ell_ 2\) so that \(d(Y,Z)\leq c_ 2 \log n\) and that (up to a possible factor of size loglog n) this is the best possible result.
For the entire collection see [Zbl 0523.00008].


46B20 Geometry and structure of normed linear spaces
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46B03 Isomorphic theory (including renorming) of Banach spaces
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