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Functional AnalysisMetric Geometrymathscidoc:1701.12027

Arkiv for Matematik, 53, (1), 1-36, 2013.3
For a compact connected set$X$⊆$ℓ$^{∞}, we define a quantity$β$′($x$,$r$) that measures how close$X$may be approximated in a ball$B$($x$,$r$) by a geodesic curve. We then show that there is$c$>0 so that if$β$′($x$,$r$)>$β$>0 for all$x$∈$X$and$r$<$r$_{0}, then $\operatorname{dim}X>1+c\beta^{2}$ . This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.
@inproceedings{jonas2013hausdorff,
title={Hausdorff dimension of wiggly metric spaces},
author={Jonas Azzam},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203639293936073},
booktitle={Arkiv for Matematik},
volume={53},
number={1},
pages={1-36},
year={2013},
}

Jonas Azzam. Hausdorff dimension of wiggly metric spaces. 2013. Vol. 53. In Arkiv for Matematik. pp.1-36. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203639293936073.