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Proceedings of GO 2005, pp. 23 – 28.The Small Octagon with Longest PerimeterCharles Audet, 1 Pierre Hansen, 2 and Frédéric Messine 31 GERAD and Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal, C.P. 6079, Succ.Centre-ville, Montréal (Québec), H3C 3A7 Canada, Charles.Audet@gerad.ca2 GERAD and Département des Méthodes Quantitatives, HEC Montréal, 3000 Chemin de la côte Sainte Catherine, MontréalH3T 2A7, Canada, École des Hautes Études Commerciales, C.P. 6079, Succ. Centre-ville, Montréal (Québec), H3C 3A7Canada, Pierre.Hansen@gerad.ca3 ENSEEIHT-IRIT, 2 rue C Camichel, 31071 Toulouse Cedex, France, Frederic.Messine@n7.frAbstractThe convex octagon with unit diameter and maximum perimeter is determined. This answers anopen question dating from 1922. The proof uses geometric reasoning and an interval arithmeticbased global optimization algorithm to solve a series of non-linear and non-convex programs involvingtrigonometric functions.This article summarizes the complete proof published in [2].1. Introduction and related problemsWe answer in this work, a geometric problem opened since 1922 by Reinhardt, [14]. We calla small n−gon a polygon with n vertices (or n edges) and with a unit diameter (the longestdistance between two extreme point is 1). In [14], Reinhardt studies the properties of maximalarea and maximal perimeter of small polygons. He proves that the regular n−gons with equalsides and equal angles have the properties to be of both maximal area and maximal perimeterif n is odd. When n is even, Reinhardt proves that the square owns the property of maximalarea and that there exists a regular polygon with only equal sides (based on a Reuleaux polygon)which has the property of maximal perimeter if n = 2 s for s ∈ IN \ {0, 1}. Therefore, theopen cases were the 4−gon for the maximal perimeter and the hexagon for the problem ofmaximal area. In 1975, Graham proves that there exists an irregular small hexagon which hasa maximal area about 4% superior to the regular one, [7]. For proving this, Graham bisectedthe global optimization problems into 10 small ones. Woodall proves in 1971, that the optimalsolutions are in this 10 configurations which are called linear thrackleations, [17]. 9 of thesecases were eliminated by using geometric arguments and the last case amounted to solve anunivariate global optimization problem which gives the optimal solution for the problem ofwhich unit diameter hexagon has the maximal area. The next open case: The largest smalloctagon was solved by Audet et al. [4], using the same way nevertheless this led to consider31 thrackleation graphs corresponding to 31 sub-problems. Therefore, one proves in [4] thatthere exists an irregular small octagon with maximal area about 2.82% greater than the regularone. This problem was extremely difficult to solve because the most difficult case (case 31which leads to the global solution about the 31 thrackleations) has 10 variables and needed 100hours of computations for a specific global optimization algorithm dedicated to non-convexquadratic programs [1].Now, considering the problem of the maximal perimeter. In 1997, Datta [6] proves thatthe small 4-gon based on a triangle of sides one with a supplementary vertex at the bisector