Metric spaces with linear extensions preserving Lipschitz condition

• Autores: Alexander Brudnyi, Yuri Brudnyi
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 129, Nº 1, 2007, págs. 217-314
• Idioma: inglés
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• Resumen

  • We study a new bi-Lipschitz invariant ?(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 ?(M). We prove that ?(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 ?(M) = 8.