83 research outputs found
Sharp distortion growth for bilipschitz extension of planar maps
This note addresses the quantitative aspect of the bilipschitz extension
problem. The main result states that any bilipschitz embedding of
into can be extended to a bilipschitz self-map of
with a linear bound on the distortion.Comment: 9 pages. Slightly expanded introduction, added reference
Symmetrization and extension of planar bi-Lipschitz maps
We show that every centrally symmetric bi-Lipschitz embedding of the circle
into the plane can be extended to a global bi-Lipschitz map of the plane with
linear bounds on the distortion. This answers a question of Daneri and Pratelli
in the special case of centrally symmetric maps. For general bi-Lipschitz
embeddings our distortion bound has a combination of linear and cubic growth,
which improves on the prior results. The proof involves a symmetrization result
for bi-Lipschitz maps which may be of independent interest.Comment: 18 pages, 3 figure
Lipschitz retraction of finite subsets of Hilbert spaces
Finite subset spaces of a metric space form a nested sequence under
natural isometric embeddings . We prove that
this sequence admits Lipschitz retractions when is a
Hilbert space.Comment: Specializes the result from v1 to Hilbert spaces. The problem remains
open for general Hadamard space
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