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

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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References in corpus (2)

• Inequalities for means of chords, with application to isoperimetric problems
• On the critical exponent in an isoperimetric inequality for chords

Cited by in corpus (1)

• On the quantitative isoperimetric inequality in the plane with the barycentric distance