This is an announcement for the paper "On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set" by Evans Harrell and Antoine Henrot.
Abstract: We consider 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 quadratic expression in the support function $h$, we show that 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 $K_1$ of finite perimeter, the set in the class we consider that is farthest away in the sense of the $L^2$ distance is always a line segment. We also prove the same property for the Hausdorff distance.
Archive classification: math.OC math.FA
Mathematics Subject Classification: 52A10; 52A40;
Remarks: 3 figures
The source file(s), HarHen1_FINALMay09.tex: 46618 bytes figure1.eps: 14493 bytes figure3.eps: 9670 bytes noyau3.eps: 10101 bytes, is(are) stored in gzipped form as 0905.1464.tar.gz with size 21kb. The corresponding postcript file has gzipped size 118kb.
Submitted from: harrell@math.gatech.edu
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http://arXiv.org/abs/0905.1464
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