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).
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
3
2
|
|
|||||
|
|
5
|
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). |
||||||||||||||
|
You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
|
0
|
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? |
||
|
|
