Tight Bounds on the Maximal Area of Small Polygons: Improved Mossinghoff Polygons

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with n = 2 m vertices is not known when m ≥ 7 . In this paper, we construct, for each n = 2 m and m ≥ 3 , a small n -gon whose area is the maximal value of a one-variable function. We show that, for all even n ≥ 6 , t...

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Bibliographic Details
Published in:Discrete & computational geometry 2023-07, Vol.70 (1), p.236-248
Main Author: Bingane, Christian
Format: Article
Language:English
Subjects:
Apexes
Codes
Combinatorics
Computational Mathematics and Numerical Analysis
Geometry
Mathematics
Mathematics and Statistics
Polygons
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Summary:A small polygon is a polygon of unit diameter. The maximal area of a small polygon with n = 2 m vertices is not known when m ≥ 7 . In this paper, we construct, for each n = 2 m and m ≥ 3 , a small n -gon whose area is the maximal value of a one-variable function. We show that, for all even n ≥ 6 , the area obtained improves by O ( 1 / n 5 ) that of the best prior small n -gon constructed by Mossinghoff. In particular, for n = 6 , the small 6-gon constructed has maximal area.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-022-00374-z