A discrete isoperimetric problem

We prove that the perimeter of any convex n-gons of diameter 1 is at most 2n sin(pi/2n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal n-gons. In fact, the convex n-gons of diameter I and perimeter 2n sin(pi/2n) are in bijective correspondence with the solutions of a diophantine problem