Tight bounds on the maximal perimeter and the maximal width of convex small polygons

Christian Bingane ()
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Christian Bingane: Polytechnique Montreal

Journal of Global Optimization, 2022, vol. 84, issue 4, No 9, 1033-1051

Abstract: Abstract A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $$n=2^s$$ n = 2 s vertices are not known when $$s \ge 4$$ s ≥ 4 . In this paper, we construct a family of convex small n-gons, $$n=2^s$$ n = 2 s and $$s\ge 3$$ s ≥ 3 , and show that the perimeters and the widths obtained cannot be improved for large n by more than $$a/n^6$$ a / n 6 and $$b/n^4$$ b / n 4 respectively, for certain positive constants a and b. In addition, assuming that a conjecture of Mossinghoff is true, we formulate the maximal perimeter problem as a nonlinear optimization problem involving trigonometric functions and, for $$n=2^s$$ n = 2 s with $$3 \le s\le 7$$ 3 ≤ s ≤ 7 , we provide global optimal solutions.

Keywords: Planar geometry; Polygons; Isodiametric problems; Maximal perimeter; Maximal width; Global optimization; 52A40; 52A10; 52B55; 90C26 (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-022-01181-9

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