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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Bibliographic Details
Published in:Journal of global optimization 2022-12, Vol.84 (4), p.1033-1051
Main Author: Bingane, Christian
Format: Article
Language:English
Subjects:
Apexes
Computer Science
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Polygons
Real Functions
Trigonometric functions
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Summary: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.
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-022-01181-9