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Extremal problems for convex polygons

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Abstract

Consider a convex polygon V n with n sides, perimeter P n , diameter D n , area A n , sum of distances between vertices S n and width W n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.

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Authors and Affiliations

  1. École Polytechnique de Montréal, Montreal, Que., Canada

    Charles Audet

  2. HEC Montréal, Montreal, Que., Canada, H3T 2A7

    Pierre Hansen

  3. ENSEEIHT-IRIT, 2 rue Camichel, 31071, Toulouse, France

    Frédéric Messine

Authors
  1. Charles Audet
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  2. Pierre Hansen
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  3. Frédéric Messine
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Correspondence to Frédéric Messine.

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Audet, C., Hansen, P. & Messine, F. Extremal problems for convex polygons. J Glob Optim 38, 163–179 (2007). https://doi.org/10.1007/s10898-006-9065-5

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