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Maximal perimeter, diameter and area of equilateral unit-width convex polygons

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Abstract

The paper answers the three distinct questions of maximizing the perimeter, diameter and area of equilateral unit-width convex polygons. The solution to each of these problems is trivially unbounded when the number of sides is even. We show that when this number is odd, the optimal solution to these three problems is identical, and arbitrarily close to a trapezoid. The paper also considers the maximization of the sum of distances between all pairs of vertices of equilateral unit-width convex polygons. Based on numerical experiments on the three first open cases, it is conjectured that the optimal solution to this fourth problem is the same trapezoid as for the three other problems.

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Authors and Affiliations

  1. GERAD and Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal, C.P. 6079, Succ. Centre-ville, Montreal, QC, H3C 3A7, Canada

    Charles Audet

  2. Institut de Mathématique de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France

    Jordan Ninin

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  1. Charles Audet
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  2. Jordan Ninin
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Correspondence to Jordan Ninin.

Additional information

Article dedicated to Pierre Hansen, pillar of the Octagon Club.

Charles Audet work was supported by NSERC grant 239436-01.

Jordan Ninin’s work was supported by the Polytechnic National Institute of Toulouse (INPT) and the French National Research Agency (ANR) through COSINUS program (project ID4CS nANR-09-COSI-005).

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Audet, C., Ninin, J. Maximal perimeter, diameter and area of equilateral unit-width convex polygons. J Glob Optim 56, 1007–1016 (2013). https://doi.org/10.1007/s10898-011-9780-4

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