The value $\frac{1}{2n}\cot(\frac{\pi}{2n})$ 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 show that the optimal width for the quadrilateral is $\frac{1}{4}\sqrt{2(3\sqrt{3}-3)}$ with a precision of 10−4. We propose two mathematical programs to determine the maximum width when n=2s with s≥3 and provide approximate, but near-optimal, solutions obtained by various heuristics and local optimization for n=8, 16, and 32.