Groupe d’études et de recherche en analyse des décisions

# Isoperimetric Polygons of Maximal Width

## Charles Audet, Pierre Hansen et Frédéric Messine

The value $\frac{1}{2n} \cot\left( \frac{\pi}{2n}\right)$ 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 $\frac{1}{4} \sqrt{ 3( 2\sqrt 3 -3 )} \approx 0.2949899\ldots$ We propose two mathematical programs to determine the maximal width when n =2s with $s\geq 3$ and provide approximate, but near-optimal, solutions obtained by various heuristics and local optimization for n =8,16 and 32.

, 24 pages