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

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 vertices are not known when s ≥ 4 . In this paper, we construct a family of convex small n -gons, n = 2 s and s ≥ 3 , and show that the perimeters and the widths obtained...

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Published in:Journal of global optimization 2022-12, Vol.84 (4), p.1033-1051
Main Author: Bingane, Christian
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Language:English
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Computer Science
Mathematics
Mathematics and Statistics
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Polygons
Real Functions
Trigonometric functions
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description 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 vertices are not known when s ≥ 4 . In this paper, we construct a family of convex small n -gons, n = 2 s and s ≥ 3 , and show that the perimeters and the widths obtained cannot be improved for large n by more than a / n 6 and 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 with 3 ≤ s ≤ 7 , we provide global optimal solutions.
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subjects Apexes
Computer Science
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Polygons
Real Functions
Trigonometric functions
title Tight bounds on the maximal perimeter and the maximal width of convex small polygons
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