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Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons

Christian Bingane(christian.bingane***at***polymtl.ca)

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

Keywords: Planar geometry, polygons, isodiametric problem, maximal area

Category 1: Nonlinear Optimization

Category 2: Global Optimization

Category 3: Combinatorial Optimization

Citation: Christian Bingane. Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons. Technical Report G-2021-58, Les cahiers du GERAD, 2021.

Download: [PDF]

Entry Submitted: 10/22/2021
Entry Accepted: 10/22/2021
Entry Last Modified: 10/22/2021

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