The $n$-diameter of planar sets of constant width

Abstract

We study the notion of $n$-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 $n$-diameter of a compact set $E$ in the plane is

$$d_n\left(E\right)=max\left(\right.\prod _{1\le i<j\le n}|z_i-z_j|{\left)\right.}^{\frac{2}{n\left(n-1\right)}},$$

where the maximum is taken over all $z_k\in E$, $k=1,2,\dots ,n$. We prove that if $n=5$, then the Reuleaux $n$-gons have the largest $n$-diameter among all sets of given constant width. The proof is based on the solution of an extremal problem for $n$-diameter.