  In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable
curves in the plane, using a multiscale sum of what is now known as Jones
$\beta$-numbers, numbers measuring flatness in a given scale and location. This
work was generalized to R^n by Okikiolu, to Hilbert space by the second author,
and has many variants in a variety of metric settings. Notably, in 2005,
Hahlomaa gave a sufficient condition for a subset of a metric space to be
contained in a rectifiable curve. We prove the sharpest possible converse to
Hahlomaa's theorem for doubling curves, and then deduce some corollaries for
subsets of metric and Banach spaces, as well as the Heisenberg group.}
