227 research outputs found
Separated nets in Euclidean space and Jacobians of biLipschitz maps
We show that there are separated nets in the Euclidean plane which are not
biLipschitz equivalent to the integer lattice. The argument is based on the
construction of a continuous function which is not the Jacobian of a
biLipschitz map.Comment: LaTeX2e file, 6 page
Characterization of the Radon-Nikodym Property in terms of inverse limits
We clarify the relation between inverse systems, the Radon-Nikodym property,
the Asymptotic Norming Property of James-Ho, and the GFDA spaces introduced in
our earlier paper on differentiability of Lipschitz maps into Banach spaces
Quasisymmetric parametrizations of two-dimensional metric spheres
We study metric spaces homeomorphic to the 2-sphere, and find conditions
under which they are quasisymmetrically homeomorphic to the standard 2-sphere.
As an application of our main theorem we show that an Ahlfors 2-regular,
linearly locally contractible metric 2-sphere is quasisymmetrically
homeomorphic to the standard 2-sphere, answering a question of Heinonen and
Semmes
Differentiable structures on metric measure spaces: A Primer
This is an exposition of the theory of differentiable structures on metric
measures spaces, in the sense of Cheeger and Keith.Comment: 23 page
Inverse limit spaces satisfying a Poincare inequality
We give conditions on Gromov-Hausdorff convergent inverse systems of metric
measure graphs (and certain higher dimensional inverse systems of metric
measure spaces) which imply that the measured Gromov-Hausdorff limit
(equivalently, the inverse limit) is a PI space, i.e. it satisfies a doubling
condition and a Poincare inequality in the sense of Heinonen-Koskela. We also
give a systematic construction of examples for which our conditions are
satisfied. Included are known examples of PI spaces, such as Laakso spaces, and
a large class of new examples. Generically our graph examples have the property
that they do not bilipschitz embed in any Banach space with Radon-Nikodym
property, but they do embed in the Banach space L_1. For Laakso spaces, these
facts were discussed in our earlier papers
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