Besicovich proved a long time ago that a straight line segment of fixed length could be rotated 360
degrees within a subset S of the Euclidean plane such that $M(S)$ is arbitrarily small-where M is two-dimensional Lebesgue measure. This solved the most general version of Kakeya's problem, which asked
how small $M(S)$ could be if no further restrictions were placed on $S$. But was it ever proved that $M(S)$
could never actually be zero? I ask because Besicovich also proved that there existed subsets T of the
Euclidean plane such that $M(T)=0$ and such that pairwise disjoint straight line segments of fixed length,
having every possible orientation, were contained in $S$ (as subsets).