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 smallwhere M is twodimensional 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).
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A Kakeya needle set cannot be of measure zero (a line segment cannot be rotated continuously within a set of measure 0). See the blog post by Terry Tao. However, there are sets of measure zero within which a line segment can be rotated by a Baire1 map (link). 


Don't you need a set of positive measure to swing a cat, even through 0.00001 radians? Intuitively you need to include a small sector of a circle to rotate a line segment about any of its points as fixed centre; and while the problem would allow you to rotate about a centre that changes, I'm not seeing how you can do without something resembling a small sector. But perhaps that is the point of what you're asking? 

