Metric Spaces with Linear Extensions Preserving Lipschitz Condition
By Y. Brudnyi, Alexander Brudnyi. 2004
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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) = ∞.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/769733b85da9506de4b3e842cc93fb1661af0b45
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Products of Lipschitz-free spaces and applications (2014, background): “ ;; Brudnyi and Brudnyi provide us with an example of a two-dimensional Riemannian manifold M, equipped with its geodesic metric, which admits a subset Fsatisfying that condition (see Theorem 2.18 in [2]). We will be particularly interested in the subset Extpt 0 (F;M) of Ext 0(F;M) consisting of the”