# A question about the Kakeya problem

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).

• Just a sketch: consider a motion of the needle (unit line segment), (assume) that it can be represented by a continuous and piece-wise $C^1$ curve in the space of Euclidean motions (translations + rotations). Restrict yourself to a $C^1$ portion such that the starting and ending angles are not the same (else there can be no total rotation). By the mean value theorem there exists a point on the path where the rate of change of the angle is non-zero. It should be straight-forward to check that the image of the needle in a neighborhood of that point contains a disk. – Willie Wong Jul 13 '10 at 15:33