## 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].

### MSC:

 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

### Keywords:

Hilbert space; metric spaces; Lipschitz map; Lipschitz norm
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