Kakeya set

Description

Let $n \ge 2$. A Kakeya set, or Besicovitch set, is a compact set $K \subseteq \R^n$ that contains a line segment of length $1$ in every direction. That is, for each unit vector $e \in \R^n$ there is a point $a \in \R^n$ with $\set{a + t e : 0 \le t \le 1} \subseteq K$. The closed ball of radius $1/2$ about the origin is a Kakeya set. For a unit vector $e$, the segment from $-e/2$ to $e/2$ lies in the ball. Besicovitch showed that there are Kakeya sets in $\R^2$ with Lebesgue measure zero. If $K \subseteq \R^2$ is such a set, then $K \times [0,1]^{n-2}$ is a Kakeya set in $\R^n$ of measure zero. Indeed, let $e = (e_1, e_2, e')$ be a unit vector. Take a segment of $K$ in the direction of $(e_1, e_2)$ with length $\abs{(e_1, e_2)}$. The last $n - 2$ coordinates then move by $te'$ for $0 \le t \le 1$, starting from a suitable corner of the cube.