Simple Polygons of Maximum Perimeter Contained in a Unit Disk

A polygon is said to be simple if the only points of the plane belonging to two of its edges are its vertices. We answer the question of finding, for a given integer n, a simple n-sided polygon contained in a disk of radius 1 that has the longest perimeter. When n is even, the optimal solution is arbitrarily close to a line segment of length 2n. When n is odd, the optimal solution is arbitrarily close to an isosceles triangle. Work of the first author was supported by NSERC grant 239436-05, AFOSR FA9550-07-1-0302, and ExxonMobil. Work of the second author was supported by NSERC grant 105574-02.