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

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 quadratic expression in the support function h, 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 L distance is always a line segment. The same property is proved for the Hausdorff distance.