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The Small Octagon with Longest Perimeter 276. Eliminating cases using IBBANow one uses IBBA in order to discard a lot of remaining tightened (by geometric arguments,see Section 3) cases corresponding to linear thrackleations.In a first step, one considers configurations which has an isolated vertex. It is a particularconfiguration such that only one vertex is above (or below by symmetry) an edge, see Figure4.•v 2v 1 v 0••Figure 4. An octagon with an isolated vertex v 1IBBA permit to eliminate all these configurations by adding the constraint that an upperbound of all the perimeters of all these configurations must be greater than p ∗ = 3.121147.Therefore, 16 cases are eliminated together, corresponding to thrackleations numbered 5, 6,7, 8, 9, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 30.Considering the remaining cases: 16, 17, 19, 20 and 31, IBBA shows in less than 5 hours foreach configurations, that all of them can have a perimeter superior to p ∗ .It remains only case of thrackleation 18, which is a relaxation of 31 and also 29 (the optimalthrackleation). In that case, the optimal conditions derived from Proposition 2 are satisfied for29, and then this problem can only be solved with a precision ɛ = 0.5 × 10 −4 . IBBA shows incase 18, that it does not exist a better solution than p ∗ + ɛ for this configuration.This completes the proof; see [2] for more details.7. ConclusionThe small octagon with longest perimeter has been determined and the length of its perimeteris equal to 3.1211 (all decimals are exact), with an error guaranteed to be less than ɛ = 0.5 ×10 −4 . As in the study of the small octagon with largest area [4], the diameter graph of anoptimal octagon must be a connected linear thrakleation. This lead to 31 cases. The optimalone is case 29 for the longest perimeter while it was case 31 for the largest area. The diametergraph for the octagon with unit-length side and smallest diameter [3] was not connected. Inall cases the optimal figure possesses an axis of symmetry. For comparison, Figure 5 illustratesthree small octagons: The regular one, the one with maximal perimeter with equal sides [3],and the one studied in this paper.The above mentioned optimal octagons were obtained by combining geometric reasoningwith global optimization algorithms [1, 10]. It appears difficult to answer any of these questionswith one of those tools alone.References[1] C. Audet, P. Hansen, B. Jaumard, and G. Savard. A branch and cut algorithm for nonconvex quadraticallyconstrained quadratic programming. Math. Program., 87(1, Ser. A):131–152, 2000.[2] C. Audet, P. Hansen, F. Messine. The small octagon with longest perimeter. Cahiers du GERAD,http://www.gerad.ca/en/publications/cahiers.php, 2005.[3] C. Audet, P. Hansen, F. Messine, and S. Perron. The minimum diameter octagon with unit-length sides:Vincze’s wife’s octagon is suboptimal. J. Combin. Theory Ser. A, 108:63–75, 2004.