MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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).

share|cite|improve this question
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

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 Baire-1 map (link).

share|cite|improve this answer
Thanks very much for providing such a complete and informative response – Garabed Gulbenkian Jul 13 '10 at 19:46
Hey, you're welcome. It wasn't a huge effort, really. – Andrey Rekalo Jul 13 '10 at 20:09
In his "Birds and frogs" Dyson attributes the following to Besicovitch: "Gentlemen. Fifty million English speak English you speak. Hundred and fifty million Russians speak English I speak." – Wadim Zudilin Jul 13 '10 at 22:44
Hi, Wadim. I speak Ukrainian English. – Andrey Rekalo Jul 14 '10 at 10:03

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?

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.