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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


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.Comment: 3 figure

Topics: Mathematics - Optimization and Control, Mathematics - Functional Analysis, 52A10, 52A40
Publisher: 'Wiley'
Year: 2009
DOI identifier: 10.1112/S0025579310000495
OAI identifier:

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