Abstract
A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with \(n=2^s\) vertices are not known when \(s \ge 4\). In this paper, we construct a family of convex small n-gons, \(n=2^s\) and \(s\ge 3\), 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 \(3 \le s\le 7\), we provide global optimal solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Audet, C., Hansen, P., Messine, F.: The small octagon with longest perimeter. J. Comb. Theory Ser. A 114(1), 135–150 (2007)
Audet, C., Hansen, P., Messine, F.: Isoperimetric polygons of maximum width. Discret. Comput. Geom. 41(1), 45–60 (2009)
Audet, C., Hansen, P., Messine, F.: Ranking small regular polygons by area and by perimeter. J. Appl. Ind. Math. 3(1), 21–27 (2009)
Audet, C., Hansen, P., Messine, F., Ninin, J.: The small octagons of maximal width. Discret. Comput. Geom. 49(3), 589–600 (2013)
Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)
Bezdek, A., Fodor, F.: On convex polygons of maximal width. Arch. Math. 74(1), 75–80 (2000)
Bingane, C.: Largest small polygons: a sequential convex optimization approach. Optim. Lett. (2022). https://doi.org/10.1007/s11590-022-01887-5
Bingane, C.: OPTIGON: extremal small polygons. https://github.com/cbingane/optigon (2020)
Datta, B.: A discrete isoperimetric problem. Geometriae Dedicata 64(1), 55–68 (1997)
Hare, K.G., Mossinghoff, M.J.: Sporadic Reinhardt polygons. Discret. Comput. Geom. 49(3), 540–557 (2013)
Hare, K.G., Mossinghoff, M.J.: Most Reinhardt polygons are sporadic. Geometriae Dedicata 198(1), 1–18 (2019)
Mossinghoff, M.J.: Isodiametric problems for polygons. Discret. Comput. Geom. 36(2), 363–379 (2006)
Mossinghoff, M.J.: An isodiametric problem for equilateral polygons. Contemp. Math. 457, 237–252 (2008)
Mossinghoff, M.J.: Enumerating isodiametric and isoperimetric polygons. J. Comb. Theory Ser. A 118(6), 1801–1815 (2011)
Reinhardt, K.: Extremale polygone gegebenen durchmessers. Jahresbericht der Deutschen Mathematiker-Vereinigung 31, 251–270 (1922)
Tamvakis, N.K.: On the perimeter and the area of the convex polygon of a given diameter. Bull. Greek Math. Soc. 28, 115–132 (1987)
Acknowledgements
The author thanks Charles Audet, Professor at Polytechnique Montreal, for helpful discussions on extremal small polygons and helpful comments on early drafts of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bingane, C. Tight bounds on the maximal perimeter and the maximal width of convex small polygons. J Glob Optim 84, 1033–1051 (2022). https://doi.org/10.1007/s10898-022-01181-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-022-01181-9