Small polygons with large area

Christian Bingane () and Michael J. Mossinghoff ()
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Christian Bingane: Polytechnique Montreal
Michael J. Mossinghoff: Center for Communications Research

Journal of Global Optimization, 2024, vol. 88, issue 4, No 9, 1035-1050

Abstract: 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.

Keywords: Polygons; Isodiametric problem; Maximal area; 52A40; 51M20; 52A38 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10898-023-01329-1

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