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), 4299-4304. https://doi.org/10.1090/proc/15051
Published inProceedings of the American Mathematical Society
DisciplineMatematiikkaGeometrinen analyysi ja matemaattinen fysiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköMathematicsGeometric Analysis and Mathematical PhysicsAnalysis and Dynamics Research (Centre of Excellence)
© 2020 American Mathematical Society
We give a sufficient condition for a general compact metric space to admit an n-rectifiable 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 n-dimensional Hausdorff measure of X, H-n (X), is finite. Suppose further that the lower n-density of the measure H-n is positive, H-n-almost everywhere in X. Then X contains an n-rectifiable subset of positive H-n-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of Csornyei-Jones.
PublisherAmerican Mathematical Society
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Related funder(s)Academy of Finland; European Commission
Funding 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.
Additional information about fundingThe first author was supported by the National Science Foundation under Grant no. NSF DMS-1758709. The second author was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). ...
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