The Kakeya needle problems asks for the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also
allowed). Surprisingly, there is no minimum area (Besicovitch 1928). Another
iterative construction which tends to as small an area as desired is called a Perron tree (Falconer 1990, Wells 1991).
When the figure is restricted to be convex, the smallest region is an equilateral triangle of unit height. Wells (1991) states that Kakeya discovered this, while Falconer (1990) attributes it to Pál.
If convexity is replaced by the weaker assumption of simply-connectedness, then the area can still be arbitrarily small, but if the set is required to be star-shaped,
then
is a known lower bound (Cunningham 1965).
The smallest simple convex domain in which one can put a segment of length 1 which will coincide with itself when rotated by has area