> Date: Wed, 14 Apr 2004 23:02:44 -0000
> From: "hartmann1052" <hartmann1052@...>
> Subject: Kakeya sets.
>
> I know that the infimum of areas of the sets in the plane that a
> needle of length 1 can be contunuously moved within the set so that
> in the end it occupies the original place but in inverted position is
> 0.
> But can we prove that there exists a set of measure zero with this
> property?If not how can we show it
>

The answer is almost certainly yes. There is a construction in

Cunningham, F., Jr.
The Kakeya problem for simply connected and for star-shaped sets.
Amer. Math. Monthly 78 1971 114--129.

of a compact set of arbitrarily small area, _and_ in a fixed ball, in which a
needle
can be continuously rotated by 180 degrees. It seems quite likely that one can
make these sets nested inside each other, and then the trajectories of the
needles
inside these sets should converge (or perhaps some subsequence will converge)
uniformly
to a continuous trajectory supported on a compact set of measure zero.

Note that the resulting trajectory will merely be continuous; it has no chance
of being continuously differentiable or even rectifiable, as such trajectories
must sweep out a nonzero amount of area by Besicovich's projection theorem.

Terry