Kakeya needle problem

In mathematics, the Kakeya needle problem asks whether there is a minimum area of a region D in the plane, in which a needle can be turned through 360°. This question was first posed by Soichi Kakeya (1886-1947), a Japanese mathematician who worked mainly in mathematical analysis. In 1917 he asked this question, about convex sets.

He seems to have suggested that D of minimum area, without the convexity restriction, would be a three-pointed astroid shape. The original problem was solved by Pal (1921). The early history of this question has been subject to some discussion, though.

Besicovitch was able to show that there is no lower bound > 0 for the area of such a region D, in which a needle of unit length can be turned round (in 1925, publication in 1928). This built on earlier work of his, on plane sets which contain a unit segment in each orientation. Such a set is now called a Besicovitch set. Besicovitch's work showing such a set could have arbitrarily small measure was from 1919. The problem may have been considered by analysts, before that.

The same question was then posed in higher dimensions. It has become part of a collection of related issues from apparently different fields, such as the Fourier transform.