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In 1917, Soichi Kakeya posed the following problem:
Kakeya needle problem. What is the least amount of area required to continuously rotate a unit line segment in the plane by a full rotation (i.e. by )?
In 1928, Besicovitch showed that given any , there exists a planar set of area at most within which a unit needle can be continuously rotated; the proof relies on the construction of what is now known as a Besicovitch set – a set of measure zero in the plane which contains a unit line segment in every direction. So the answer to the Kakeya needle problem is “zero”.
I was recently asked (by Claus Dollinger) whether one can take ; in other words,
Question. Does there exist a set of measure zero within which a unit line segment can be continuously rotated by a full rotation?
This question does not seem to be explicitly answered in the literature. In the papers of von Alphen and of Cunningham, it is shown that it is possible to continuously rotate a unit line segment inside a set of arbitrarily small measure and of uniformly bounded diameter; this result is of course implied by a positive answer to the above question (since continuous functions on compact sets are bounded), but the converse is not true.
Below the fold, I give the answer to the problem… but perhaps readers may wish to make a guess as to what the answer is first before proceeding, to see how good their real analysis intuition is. (To partially prevent spoilers for those reading this post via RSS, I will be whitening the text; you will have to highlight the text in order to see it. Unfortunately, I do not know how to white out the LaTeX in such a way that it is visible upon highlighting, so RSS readers may wish to stop reading right now; but I suppose one can view the LaTeX as supplying hints to the problem, without giving away the full solution.)
[Update, March 13: a non-whitened version of this article can be found as part of this book.]