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

TL;DR

This paper studies the maximization of certain quadratic functionals over convex sets with fixed perimeter and Steiner point, showing that extremal solutions are either triangles or line segments, with implications for farthest convex sets.

Contribution

It characterizes the maximizers of quadratic integral functionals on convex sets, proving they are either triangles or line segments, and applies this to identify farthest convex bodies.

Findings

Maximizers are triangles or line segments.

Farthest convex set in L^2 distance is a line segment.

Farthest convex set in Hausdorff distance is also a line segment.

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.