Abstract
A polygon is small if it has unit diameter. The maximal area of a small polygon with a fixed number of sides n is not known when n is even and \(n\ge 14\). We determine an improved lower bound for the maximal area of a small n-gon for this case. The improvement affects the \(1/n^3\) term of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be improved by more than \(O(1/n^3)\). For \(n=6\), 8, 10, and 12, the polygon we construct has maximal area.
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Audet, C., Hansen, P., Messine, F., Xiong, J.: The largest small octagon. J. Combin. Theory Ser. A 98(1), 46–59 (2002). https://doi.org/10.1006/jcta.2001.3225
Audet, C., Hansen, P., Messine, F.: Extremal problems for convex polygons. J. Glob. Optim. 38(2), 163–179 (2007). https://doi.org/10.1007/s10898-006-9065-5
Audet, C., Hansen, P., Messine, F.: The small octagon with longest perimeter. J. Combin. Theory Ser. A 114(1), 135–150 (2007). https://doi.org/10.1016/j.jcta.2006.04.002
Audet, C., Hansen, P., Messine, F.: Extremal problems for convex polygons–an update. Lectures on Global Optimization, Fields Inst. Comm. Amer. Math. Soc. 55, 1–16 (2009). https://doi.org/10.1090/fic/055/01
Audet, C., Hansen, P., Svrtan, D.: Using symbolic calculations to determine largest small polygons. J. Glob. Optim. 81(1), 261–268 (2021). https://doi.org/10.1007/s10898-020-00908-w
Bieri, H.: Ungelöste probleme: Zweiter nachtrag zu nr. 12. Elem. Math. 16, 105–106 (1961)
Bingane, C.: OPTIGON: extremal small polygons (2022), https://github.com/cbingane/ optigon
Bingane, C.: Tight bounds on the maximal perimeter and the maximal width of convex small polygons. J. Glob. Optim. 84(4), 1033–1051 (2022). https://doi.org/10.1007/s10898-022-01181-9
Bingane, C.: Largest small polygons: a sequential convex optimization approach. Optim. Lett. 17(2), 385–397 (2023). https://doi.org/10.1007/s11590-022-01887-5
Bingane, C.: Tight bounds on the maximal area of small polygons: improved Mossinghoff polygons. Discret. Comput. Geom. 70(1), 236–248 (2023). https://doi.org/10.1007/s00454-022-00374-z
Foster, J., Szabo, T.: Diameter graphs of polygons and the proof of a conjecture of Graham. J. Combin. Theory Ser. A 114(8), 1515–1525 (2007). https://doi.org/10.1016/j.jcta.2007.02.006
Graham, R.L.: The largest small hexagon. J. Combin. Theory Ser. A 18, 165–170 (1975). https://doi.org/10.1016/0097-3165(75)90004-7
Henrion, D., Messine, F.: Finding largest small polygons with GloptiPoly. J. Glob. Optim. 56(3), 1017–1028 (2013). https://doi.org/10.1007/s10898-011-9818-7
Mossinghoff, M.J.: Isodiametric problems for polygons. Discret. Comput. Geom. 36(2), 363–379 (2006). https://doi.org/10.1007/s00454-006-1238-y
Pintér, J.D.: Largest small \(n\)-polygons: numerical optimum estimates for \(n \ge 6\). Num. Anal. Optim. 354, 231–247 (2020). https://doi.org/10.1007/978-3-030-72040-711
Pintér, J.D., Kampas, F.J., Castillo, I.: Finding the sequence of largest small \(n\)-polygons by numerical optimization. Math. Comput. Appl. 27(3), 10 (2022). https://doi.org/10.3390/mca27030042
Reinhardt, K.: Extremale Polygone gegebenen Durchmessers. Jahresber. Deutsch. Math. Verein. 31, 251–270 (1922)
Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Progr. 106, 25–57 (2006). https://doi.org/10.1007/s10107-004-0559-y
Yuan, B.: The Largest Small Hexagon. National University of Singapore (2004)
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Bingane, C., Mossinghoff, M.J. Small polygons with large area. J Glob Optim 88, 1035–1050 (2024). https://doi.org/10.1007/s10898-023-01329-1
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DOI: https://doi.org/10.1007/s10898-023-01329-1