The value is shown to be an upper bound on the width of any n-sided polygon with unit perimeter. This bound is reached when n is not a power of 2, and the corresponding optimal solutions are the regular polygons when n is odd, and clipped regular Reuleaux polygons when n is even but not a power of 2. Using a global optimization algorithm, we solve the problem for n =4. The optimal width for the quadrilateral is shown to be We propose two mathematical programs to determine the maximal width when n =2^s with and provide approximate, but near-optimal, solutions obtained by various heuristics and local optimization for n =8,16 and 32.