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arXiv · 0905.1464

On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set

Evans Harrell·Antoine Henrot
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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.

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Evans Harrell, Antoine Henrot. 2009-05-10. On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set. https://doi.org/10.1112/s0025579310000495

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