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 $360^\circ$)?

In 1928, Besicovitch showed that given any $\varepsilon > 0$, there exists a planar set of area at most $\varepsilon$ 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 $\varepsilon = 0$; 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.]

— Solution —

The answer to the question is … no.

To see this, let $E \subset {\Bbb R}^2$ be a set in the plane within which a unit line segment can be continuously rotated.  This means that there exists a continuous map $l: t \mapsto l(t)$ from times $t \in [0,1]$ to unit line segments $l(t) \subset E$.  We can parameterise each such line segment as

$l(t) = \{ (x(t) + s \cos \omega(t),y(t) + s \sin \omega(t) ): -0.5 \leq s \leq 0.5 \}$

where $x, y, \omega: [0,1] \to {\Bbb R}$ are continuous functions.

to disguise the length of the proof,

and also because from this point onwards,

it’s pretty obvious that

we are proving that

the answer to the question is negative,

rather than trying to build an example. ]

Recall that on a compact set, all continuous functions are uniformly continuous. In particular, there exists $\varepsilon > 0$ such that

$|x(t)-x(t')|, |y(t)-y(t')|, |\omega(t)-\omega(t')| \leq 0.001$ (1)

(say) whenever $t, t' \in [0,1]$ are such that $|t-t'| \leq \varepsilon$.

Fix this $\varepsilon$.  Observe that $\omega(t)$ cannot be a constant function of t, otherwise the needle would never rotate.  We conclude that there must exist $t_0, t_1 \in [0,1]$ with $|t_0-t_1| \leq \varepsilon$ and $\omega(t_0) \neq \omega(t_1)$.

Without loss of generality, we may assume that $t_0 < t_1$ and $x(t_0)=y(t_0)=\omega(t_0)=0$.

Now let a be any real number between -0.4 and +0.4.  From (1) we see that for any $t_0 \leq t \leq t_1$, the line l(t) intersects the line x=a in some point $(a, y_a(t))$, which must therefore lie in E.  Furthermore, $y_a(t)$ varies continuously in t.  By the intermediate value theorem, we conclude that the interval between $(a,y_a(t_0))$ and $(a,y_a(t_1))$ lies in E.  Taking unions over all a between -0.4 and +0.4, we see that E contains a non-trivial sector, and thus has non-zero area.  The claim follows.
Remark. A variant of this argument shows a stronger statement, namely that for any fixed c>0, any set E whose measure is sufficiently small (depending on c) within which a unit line segment can be rotated by at least c, must have a diameter of at least 2-c.  (A similar point was already made in the Cunningham paper referenced above.)  $\diamond$