The Kakeya problem seeks the smallest convex region in which a unit segment can be moved back to itself but in the opposite direction. The answer is an equilateral triangle of unit height [3].

If a nonconvex region is allowed, an area less than that of the deltoid is possible.

This Demonstration shows a simply connected Kakeya set constructed by Cunningham that is the first step in constructing Kakeya sets of smaller area.

Given two parallel lines, a Pál joint lets you move a unit line segment continuously from one to the other through a set of arbitrarily small area.

A Kakeya set (or Besicovitch set) is one that can contain a unit segment in any direction.

A Kakeya needle set is a set through which a line segment can be moved continuously back to itself but turned 180°. (Thus a Kakeya needle set is a Kakeya set.)

Cunningham [7] proved the following theorem:

Given , there exists a simply connected Kakeya set of area less than contained in a circle of radius 1.

See Besicovitch's talk [5] and expository article [6].