Isodiametric Problems for Polygons

The maximal area of a polygon with n = 2m edges and unit diameter is not known when m greater than or equal to 5, nor is the maximal perimeter of a convex polygon with n = 2m edges and unit diameter known when m greater than or equal to 4. We construct improved polygons in both problems, and show that the values we obtain cannot be improved for large n by more than c1/n3 in the area problem and c2/n5 in the perimeter problem, for certain constants c1 and c2. [PUBLICATION ABSTRACT]