Metric Spaces with Linear Extensions Preserving Lipschitz Condition
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 importa...
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| Veröffentlicht in: | American journal of mathematics 2007-02, Vol.129 (1), p.217-314 |
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| creator | Brudnyi, Alexander Brudnyi, Yuri |
| description | 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) = ∞. |
| doi_str_mv | 10.1353/ajm.2007.0000 |
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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. 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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) = ∞.</description><subject>Banach space</subject><subject>Coordinate systems</subject><subject>Curvature</subject><subject>Exact sciences and technology</subject><subject>Functional analysis</subject><subject>General topology</subject><subject>Geometry</subject><subject>Linear transformations</subject><subject>Mathematical analysis</subject><subject>Mathematical functions</subject><subject>Mathematical problems</subject><subject>Mathematical theorems</subject><subject>Mathematics</subject><subject>Real functions</subject><subject>Riemann manifold</subject><subject>Sciences and techniques of general use</subject><subject>Topology. Manifolds and cell complexes. 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| subjects | Banach space Coordinate systems Curvature Exact sciences and technology Functional analysis General topology Geometry Linear transformations Mathematical analysis Mathematical functions Mathematical problems Mathematical theorems Mathematics Real functions Riemann manifold Sciences and techniques of general use Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds Vertices |
| title | Metric Spaces with Linear Extensions Preserving Lipschitz Condition |
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