TITLE: A question about the Osgood curve
QUESTION [9 upvotes]: Does every sub-arc of the Osgood curve (with distinct end-points) have positive two-dimensional 
Lebesgue measure? If not, do there exist Jordan curves whch have this property?

REPLY [2 votes]: Another example: in a 1961 paper, Besicovitch and Schoenberg constructed Jordan arcs with no negligible subarcs. More precisely, according to (1), for every $\varepsilon > 0$, there exists a Jordan arc $J$ such that for $v \neq v'$ in $J$, denoting $J(v,v')$ the subarc having $v$ and $v'$ as endpoints, $$0 < m_2\big(J(v,v')\big) < C_{\varepsilon} |v'-v|^{2-\varepsilon}.$$
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References 

$\ \ $ (1)$\ \ $ On Jordan Arcs and Lipschitz Classes of Functions Defined on Them