ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2009, Vol. 3, No. 1, pp. 21–27. c  Pleiades Publishing, Ltd., 2009. Original Russian Text c  Ch. Audet, P. Hansen, F. Messine, 2008, published in Diskretnyi Analiz i Issledovanie Operatsii, 2008, Vol. 15, No. 3, pp. 65–73. Ranking Small Regular Polygons by Area and by Perimeter Ch. Audet 1* , P. Hansen 2** , and F. Messine 3*** 1 GERAD and D ´ epartement de Math ´ ematiques et de G ´ enie Industriel, ´ Ecole Polytechnique de Montr ´ eal, C.P. 6079, Succ. Centre-ville, Montr ´ eal (Qu ´ ebec), H3C 3A7 Canada 2 GERAD and D ´ epartement des M ´ ethodes Quantitatives de Gestion, HEC Montr ´ eal, 3000 Chemin de la c ˆ ote Sainte Catherine, Montr ´ eal H3T 2A7, Canada, ´ Ecole des Hautes ´ Etudes Commerciales, C.P. 6079, Succ. Centre-ville, Montr ´ eal (Qu ´ ebec), H3C 3A7 Canada 3 ENSEEIHT-IRIT, UMR-CNRS 5828, 2 rue Camichel, 31000 Toulouse, France Received October 10, 2007; in ﬁnal form, March 3,2008 Abstract—From the pentagon onwards, the area of the regular convex polygon with n sides and unit diameter is greater for each odd number n than for the next even number n +1. Moreover, from the heptagon onwards, the diﬀerence in areas decreases as n increases. Similar properties hold for the perimeter. A new proof of a result by K. Reinhardt follows. DOI: 10.1134/S1990478909010037 1. INTRODUCTION Extremal problems on convex polygons have been studied since the ancient Greeks (see [3, 5, 6, 9, 15, 16] for surveys). The three following problems are central to this stream of research: P 1 : What is the maximum area of an n-sided polygon with ﬁxed perimeter? P 2 : What is the maximum area of an n-sided polygon with ﬁxed diameter? P 3 : What is the maximum perimeter of a convex n-sided polygon with ﬁxed diameter? Problem P 1 was solved by Zenodorus in the second century b.c.e. (implicitly assuming the existence of a solution): The regular n-sided polygon has maximal area for all n. His proof, as well as numerous further ones using a variety of tools, is presented in a beautiful paper by V. Bl ˚ asj ¨ o [5]. For Problem P 2 , a fundamental result was obtained by K. Reinhardt in 1922 [17]: For odd n, the regular n-gon has maximal area among all isodiametric n-gons. Moreover, for every even n at least 6, the regular n-gon does not have maximum area. R. L. Graham [13] determined the hexagon with a ﬁxed diameter and maximum area, which is about 3.92% above the area of the regular hexagon. Combining the geometric reasoning with the extensive use of an algorithm for nonconvex quadratic program- ming [1], the extremal octagon, which has an area about 2.79% above that of the regular octagon, was found in [4]. Several authors obtained, with nonlinear programming codes, e.g., LANCELOT [8] and SNOPT [12], some heuristic solutions for n even and at least 10. The numerical experiments from [7, 10, 11] are summarized in [3]. To bound the error, M. J. Mossinghoﬀ [15, 16] focussed on the n-gons with a particular diameter conﬁguration introduced in [13]; i.e., an (n − 1)-star polygon with a pending edge. Then, assuming the axial symmetry and the equality of some angles between diameters, the n-gons are obtained with an area very close to be optimal. Indeed, it is shown that the areas obtained cannot be improved for large n by more than c 1 /n 3 , where c 1 is a constant. For Problem P 3 , K. Reinhardt [17] proved that the regular n-gons have extremal perimeter in the same cases when they have the extremal area: For odd n, the regular n-gon has the maximum perimeter among all isodiametric n-gons. Moreover, if n has an odd factor m ≥ 3 then the n-gon with a maximum perimeter can be obtained as follows: * E-mail: Charles.Audet@gerad.ca ** E-mail: Pierre.Hansen@gerad.ca *** E-mail: Frederic.Messine@n7.fr 21