Extremal Problems for Convex Polygons ∗ Charles Audet ´ Ecole Polytechnique de Montr´ eal Pierre Hansen HEC Montr´ eal Fr´ ed´ eric Messine ENSEEIHT-IRIT Abstract. Consider a convex polygon V n with n sides, perimeter P n , diameter D n , area A n , sum of distances between vertices S n and width W n . Minimizing or maximizing any of these quantities while fixing another defines ten pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization meth- ods. Numerous open problems are mentioned, as well as series of test problems for global optimization and nonlinear programming codes. Keywords: polygon, perimeter, diameter, area, sum of distances, width, isoperimeter problem, isodiametric problem. 1. Introduction Plane geometry is replete with extremal problems, many of which are de- scribed in the book of Croft, Falconer and Guy [12] on Unsolved problems in geometry. Traditionally, such problems have been solved, some since the Greeks, by geometrical reasoning. In the last four decades, this approach has been increasingly complemented by global optimization methods. This allowed solution of larger instances than could be solved by any one of these two approaches alone. Probably the best known type of such problems are circle packing ones: given a geometrical form such as a unit square, a unit-side triangle or a unit- diameter circle, find the maximum radius and configuration of n circles which can be packed in its interior (see [46] for a recent survey and the site [44] for a census of exact and approximate results with up to 300 circles). Extremal problems on convex polygons have also attracted attention of both geometers and optimizers. In this paper, we survey research on that topic. A polygon V n is a closed plane figure with n sides. A vertex of V n is a point at which two sides meet. If any line segment joining two points of V n is entirely within V n then V n is convex. If all sides of V n have equal length, V n is equilateral. If an equilateral polygon has equal inner angles between ∗ Work of the first author was supported by NSERC grant 239436-01 and AFOSR grant F49620-01-1-0013. Work of the second author was supported by NSERC grant 105574-02. c  2005 Kluwer Academic Publishers. Printed in the Netherlands. extremal.tex; 22/12/2005; 15:09; p.1