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 .
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  proved the following theorem:
, there exists a simply connected Kakeya set of area less than
contained in a circle of radius 1.
See Besicovitch's talk  and expository article .
 K. J. Falconer, The Geometry of Fractal Sets
, 1st ed., Cambridge: Cambridge University Press, 1990.
 D. Wells, The Penguin Dictionary of Curious and Interesting Geometry
, London: Penguin Books, 1991.
 A. S. Besicovitch, "The Kakeya Problem," The American Mathematical Monthly, 70
, 1963 pp. 697–706.