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The largest small octagon. (English) Zbl 1022.90013

Summary: Thrackleation of graphs and global optimization for quadratically constrained quadratic programming are used to find the octagon with unit diameter and largest area. This proves the first open case of a conjecture of R. L. Graham [ibid. 18, 165-170 (1975; Zbl 0299.52006)].

MSC:

90C20 Quadratic programming
51-XX Geometry
52-XX Convex and discrete geometry

Citations:

Zbl 0299.52006
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References:

[1] Audet, C., Optimisation globale structurée: propriétés, équivalences et résolution (1997), École Polytechnique de Montréal
[2] Audet, C.; Hansen, P.; Jaumard, B.; Savard, G., A branch and cut algorithm for nonconvex quadratically constrained quadratic programming, Math. Programming, 87, 131-152 (2000) · Zbl 0966.90057
[3] Graham, R. L., The largest small hexagon, J. Combin. Theory Ser. A, 18, 165-170 (1975) · Zbl 0299.52006
[4] Reinhardt, K., Extremale Polygone gegebenen Durchmessers, Jahresber. Deutsch. Math. Verein, 31, 251-270 (1922)
[5] Woodall, D. R., Thrackles and deadlock, (Welsh, D. J.A, Combinatorial Mathematics and Its Applications (1971), Academic Press: Academic Press New York) · Zbl 0213.50603
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