Skip to main content Accessibility help
×
Home

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

  • Evans M. Harrell II (a1) and Antoine Henrot (a2)

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.

Copyright

References

Hide All
[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), 16.
[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), 225233; Addendum, Ibid., 77 (2006), 219.
[3]Groemer, H., Geometric Applications of Fourier Series and Spherical Harmonics (Encyclopedia of Mathematics and its Applications 61), Cambridge University Press (Cambridge, 1996).
[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), 301318.
[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).
[6]Henrot, A. and Pierre, M., Variation et optimisation de formes (Mathématiques et Applications 48), Springer (Berlin, 2005).
[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), 153176.
[8]Lamboley, J. and Novruzi, A., Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim. 48(5) (2009), 30033025.
[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), 98110.
[10]McClure, D. E. and Vitale, R. A., Polygonal approximation of plane convex bodies. J. Math. Anal. Appl. 51 (1975), 326358.
[11]McMullen, P., The Hausdorff distance between compact convex sets. Mathematika 31 (1984), 7682.
[12]Pólya, G. and Szegő, G., Isoperimetric Inequalities in Mathematical Physics (Annals of Mathematics Studies AM-27), Princeton University Press (Princeton, 1951).
[13]Rockafellar, R. T., Convex Analysis, Princeton University Press (Princeton, 1970).
[14]Schneider, R., Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia of Mathematics and its Applications 44), Cambridge University Press (Cambridge, 1993).
[15]Webster, R., Convexity, Oxford University Press (Oxford, 1994).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed