To any metric space it is possible to associate
the cardinal invariant corresponding to the least number of
rectifiable curves in the space whose union is not meagre. It is
shown that this invariant can vary with the metric space considered,
even when restricted to the class of convex subspaces of separable
Banach spaces. As a corollary it is obtained that it is consistent
with set theory that that any set of reals of size \aleph_1 is
meagre yet there are \aleph_1 rectifiable curves in **R**^{3}
whose union is not meagre. The consistency of this statement when
the phrase "rectifiable curves" is replaced by "straight lines"
remains open.

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