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Small polygons with large area

Author

Listed:
  • Christian Bingane

    (Polytechnique Montreal)

  • Michael J. Mossinghoff

    (Center for Communications Research)

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 $$n\ge 14$$ n ≥ 14 . We determine an improved lower bound for the maximal area of a small n-gon for this case. The improvement affects the $$1/n^3$$ 1 / n 3 term of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be improved by more than $$O(1/n^3)$$ O ( 1 / n 3 ) . For $$n=6$$ n = 6 , 8, 10, and 12, the polygon we construct has maximal area.

Suggested Citation

  • Christian Bingane & Michael J. Mossinghoff, 2024. "Small polygons with large area," Journal of Global Optimization, Springer, vol. 88(4), pages 1035-1050, April.
    • Handle: RePEc:spr:jglopt:v:88:y:2024:i:4:d:10.1007_s10898-023-01329-1
    DOI: 10.1007/s10898-023-01329-1
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    References listed on IDEAS

    as
    1. Christian Bingane, 2022. "Tight bounds on the maximal perimeter and the maximal width of convex small polygons," Journal of Global Optimization, Springer, vol. 84(4), pages 1033-1051, December.
    2. Didier Henrion & Frédéric Messine, 2013. "Finding largest small polygons with GloptiPoly," Journal of Global Optimization, Springer, vol. 56(3), pages 1017-1028, July.
    3. Charles Audet & Pierre Hansen & Dragutin Svrtan, 2021. "Using symbolic calculations to determine largest small polygons," Journal of Global Optimization, Springer, vol. 81(1), pages 261-268, September.
    Full references (including those not matched with items on IDEAS)

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