The Kakeya needle problem asks for the plane figure of least area in which a line segment
of length 1 can be freely rotated (where translation
of the line 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 a convex set, 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 the requirement of a convex set is replaced by the weaker assumption of being simply connected,
then the area can still be arbitrarily small, but if the
set is required to be star convex,
then
is a known lower bound (Cunningham 1971).
Cunningham and Schoenberg (1965) construct star convex domains in which a unit line segment can be rotated,
with areas approaching
(OEIS A093823; Le Lionnais 1983). This gives an upper bound for the infimum of the areas
of star convex domains with this property.
is a known lower bound (Cunningham 1971).
