The Kakeya Problem

Suppose a subset of Euclidean space contains a unit segment pointing in every direction. How small can the subset be?

It turns out there exist such sets, called Besicovitch sets, of Lebesgue measure zero. However, in the plane, a Besicovitch set must at least have full Hausdorff dimension.

The question for the minimal Hausdorff dimension of a Besicovitch set in higher dimensions remains one of the main open problems in harmonic analysis.