On convex polygons of maximal width

In this paper we consider the problem of finding the n-sided (n greater than or equal to 3) polygons of diameter 1 which have the largest possible width w(n). We prove that w(4) = w(3) = root 3/2 and, in general, w(n) less than or equal to cos pi/2n. Equality holds if n has an odd divisor greater than 1 and in this case a polygon 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 P.