Hausdorff dimension of wiggly metric spaces

For a compact connected set $X\subseteq \ell^{\infty}$, 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 there is $c>0$ so that if $β'(x,r)>β>0$ for all $x\in X$ and $r<r_{0}$, then $\dim X>1+cβ^{2}$. This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.

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