We pose the following problem:
(*) Does there exist a compact collection of pairwise disjoint lines
(in the space of all lines in \(R^n\)) that contains lines in every
direction?
Although no dimension (or any other quantitative parameter) appears in this problem, it is still strongly connected to the Kakeya problem: it
turns out that a negative answer would imply that in \(R^n\) any
Besicovitch set must have Hausdorff dimension at least \(n-1\).
If we relax "compact" to "closed" in (*), then we can give a construction.
Joint work with Attila Gáspár and András Máthé.