A note on topological dimension, Hausdorff measure, and rectifiability
David, G. C., & Le Donne, E. (2020). A note on topological dimension, Hausdorff measure, and rectifiability. Proceedings of the American Mathematical Society, 148(10), 42994304. https://doi.org/10.1090/proc/15051
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Proceedings of the American Mathematical SocietyDate
2020Discipline
MatematiikkaGeometrinen analyysi ja matemaattinen fysiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköMathematicsGeometric Analysis and Mathematical PhysicsAnalysis and Dynamics Research (Centre of Excellence)Copyright
© 2020 American Mathematical Society
We give a sufficient condition for a general compact metric space to admit an nrectifiable piece, as a consequence of a recent result of David Bate. Let X be a compact metric space of topological dimension n. Suppose that the ndimensional Hausdorff measure of X, Hn (X), is finite. Suppose further that the lower ndensity of the measure Hn is positive, Hnalmost everywhere in X. Then X contains an nrectifiable subset of positive Hnmeasure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of CsornyeiJones.
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https://converis.jyu.fi/converis/portal/detail/Publication/42016805
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Academy of Finland; European CommissionFunding program(s)
Academy Research Fellow, AoF; Academy Project, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
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The first author was supported by the National Science Foundation under Grant no. NSF DMS1758709. The second author was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘SubRiemannian Geometry via Metricgeometry and Liegroup Theory’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). ...License
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