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Simultaneous Lipschitz extensions. (English) Zbl 1133.26300

Russ. Math. Surv. 60, No. 6, 1057-1076 (2005); translation from Usp. Mat. Nauk 60, No. 6, 53-72 (2005).
Summary: 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 \(\lambda(M) = \infty\).

MSC:

26A16 Lipschitz (Hölder) classes
26B35 Special properties of functions of several variables, Hölder conditions, etc.
54E35 Metric spaces, metrizability
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46B99 Normed linear spaces and Banach spaces; Banach lattices
54C20 Extension of maps
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