We study the notion of
-diameter
for sets of constant width. A convex set in the plane is said to be of
constant width if the distance between two parallel support lines is
constant, independent of the direction. The Reuleaux triangles are the
well-known examples of sets of constant width that are not disks. The
-diameter of a
compact set
in the plane is
where the maximum is taken over all
,
. We prove that if
, then the Reuleaux
-gons have the
largest
-diameter
among all sets of given constant width. The proof is based on the solution of an extremal problem
for
-diameter.
Keywords
$n$-diameter, constant width sets, Pólya extremal problem