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ON THE MAXIMIZATION OF A CLASS OF FUNCTIONALS ON CONVEX REGIONS, AND THE CHARACTERIZATION OF THE FARTHEST CONVEX SET

Part of: Polytopes and polyhedra Manifolds General convexity

Published online by Cambridge University Press:  18 May 2010

Evans M. Harrell II  and
Antoine Henrot
Evans M. Harrell II
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, U.S.A. (email: harrell@math.gatech.edu)
Antoine Henrot
Affiliation:
Institut Élie Cartan Nancy, UMR 7502, Nancy Université—CNRS—INRIA, B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France (email: henrot@iecn.u-nancy.fr)
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Abstract

This article considers a family of functionals J to be maximized over the planar convex sets K for which the perimeter and Steiner point have been fixed. Assuming that J is the integral of a positive quadratic expression in the support function h and its derivative, the maximizer is always either a triangle or a line segment (which can be considered as a collapsed triangle). Among the concrete consequences of the main theorem is the fact that, given any convex body K1 of finite perimeter, the set in this class that is farthest away in the sense of the L2 distance is always a line segment. The same property is proved for the Hausdorff distance.

MSC classification

Secondary: 52A40: Inequalities and extremum problems 52A10: Convex sets in $2$ dimensions (including convex curves) 52B60: Isoperimetric problems for polytopes 49Q10: Optimization of shapes other than minimal surfaces
Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010 , pp. 245 - 265
Copyright
Copyright © University College London 2010

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References

[1]Exner, P., Fraas, M. and Harrell, E. M. II, On the critical exponent in an isoperimetric inequality for chords. Phys. Lett. A 368 (2007), 1–6.CrossRefGoogle Scholar
[2]Exner, P., Harrell, E. M. II and Loss, M., Inequalities for means of chords, with application to isoperimetric problems. Lett. Math. Phys. 75 (2006), 225–233; Addendum, Ibid., 77 (2006), 219.CrossRefGoogle Scholar
[3]Groemer, H., Geometric Applications of Fourier Series and Spherical Harmonics (Encyclopedia of Mathematics and its Applications 61), Cambridge University Press (Cambridge, 1996).CrossRefGoogle Scholar
[4]Gruber, P. M., The space of convex bodies. In Handbook of Convex Geometry, (eds Gruber, P. M. and Wills, J. M.), Elsevier (Amsterdam, 1993), 301–318.CrossRefGoogle Scholar
[5]Harrell, E. M. II and Henrot, A., On the maximum of a class of functionals on convex regions, and the means of chords weighted by curvature (in preparation).Google Scholar
[6]Henrot, A. and Pierre, M., Variation et optimisation de formes (Mathématiques et Applications 48), Springer (Berlin, 2005).CrossRefGoogle Scholar
[7]Lachand-Robert, T. and Peletier, M. A., Newton’s problem of the body of minimal resistance in the class of convex developable functions. Math. Nachr. 226 (2001), 153–176.3.0.CO;2-2>CrossRefGoogle Scholar
[8]Lamboley, J. and Novruzi, A., Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim. 48(5) (2009), 3003–3025.CrossRefGoogle Scholar
[9]Maurer, H. and Zowe, J., First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Program. 16(1) (1979), 98–110.CrossRefGoogle Scholar
[10]McClure, D. E. and Vitale, R. A., Polygonal approximation of plane convex bodies. J. Math. Anal. Appl. 51 (1975), 326–358.CrossRefGoogle Scholar
[11]McMullen, P., The Hausdorff distance between compact convex sets. Mathematika 31 (1984), 76–82.CrossRefGoogle Scholar
[12]Pólya, G. and Szegő, G., Isoperimetric Inequalities in Mathematical Physics (Annals of Mathematics Studies AM-27), Princeton University Press (Princeton, 1951).Google Scholar
[13]Rockafellar, R. T., Convex Analysis, Princeton University Press (Princeton, 1970).CrossRefGoogle Scholar
[14]Schneider, R., Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia of Mathematics and its Applications 44), Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
[15]Webster, R., Convexity, Oxford University Press (Oxford, 1994).CrossRefGoogle Scholar