Simultaneous Lipschitz extensions

This paper is devoted to a study of a new bi-Lipschitz invariant $\lambda(M)$ of metric spaces $M$. Finiteness of this quantity means that the Lipschitz functions on any subset of $M$ can be linearly extended to functions on $M$ with Lipschitz constants increased by the factor $\lambda(M)$. It is shown that $\lambda(M)$ is finite for some important classes of metric spaces, including metric trees of any cardinality, groups of polynomial growth, hyperbolic groups in the Gromov sense, certain classes of Riemannian manifolds of bounded geometry, and finite direct sums of any combinations of these objects. On the other hand, an example is given of a two-dimensional Riemannian manifold $M$ of bounded geometry with $\lambda(M)=\infty$.