Use this tag for questions relating to Kakeya and Besicovitch sets
Let $E$ be a subset of $\mathbb{R}^n$ with a unit line degment in every direction. This is a Kakeya set.
They have arbitrarily small Lebesgue measure and imply the solution to the Kakeya needle problem: What is the minimum area in the plane needed to rotate a unit line segment $360^o$?
The conjecture is trivial in one dimension, settled in two dimensions, open in dimensions $3$ and higher.
It has been connected to the areas: combinatorial geometry, additive combinatorics, multiscale analysis, heat flows, algebraic geometry, and algebraic topology.