### G-2007-56

# Simple Polygons of Maximum Perimeter Contained in a Unit Disk

## Charles Audet, Pierre Hansen, and Frédéric Messine

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*, the 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.

Published **July 2007**
,
13 pages

This cahier was revised in **May 2008**