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Abstract. We prove that the trace of the space C 1,ω (R n ) to an arbitrary closed subset X ⊂ R n is characterized by the following "finiteness" property. A function f : X → R belongs to the trace space if and only if the restriction f | Y to an arbitrary subset Y ⊂ X consisting of at most 3·2 n−1 can be extendedThe constant 3 · 2 n−1 is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.

We study a new bi-Lipschitz invariant λ(M ) of a metric space M ; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are expanded by a factor controlled by λ(M ). We prove that λ(M ) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain classes of Riemannian manifolds of bounded geometry and the finite direct sums of arbitrary combinations of these objects. On the other hand we construct an example of a two-dimensional Riemannian manifold M of bounded geometry for which λ(M ) = ∞. * Research supported in part by NSERC. 2000 Mathematics Subject Classification. Primary 26B35, Secondary 54E35, 46B15.