On convex polygons of maximal width

Abstract.

In this paper we consider the problem of finding the n-sided ($n\geq 3$) polygons of diameter 1 which have the largest possible width w _n . We prove that $w_4=w_3= {\sqrt 3 \over 2}$ and, in general, $w_n \leq \cos {\pi \over 2n}$. Equality holds if n has an odd divisor greater than 1 and in this case a polygon $\cal P$ is extremal if and only if it has equal sides and it is inscribed in a Reuleaux polygon of constant width 1, such that the vertices of the Reuleaux polygon are also vertices of $\cal P$.