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

Evans M. Harrell (1) , Antoine Henrot (2, 3)

1 School of Mathematics - Georgia Institute of Technology
2 IECN - Institut Élie Cartan de Nancy
3 CORIDA - Robust control of infinite dimensional systems and applications

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 $K_1$ of finite perimeter, the set in this class that is farthest away in the sense of the $L^2$ distance is always a line segment. The same property is proved for the Hausdorff distance.

Mots clés: isoperimetric problem; shape optimization; convex geometry; polygons; farthest convex set

Domaines: Géométrie métrique [math.MG]

Evans M. Harrell, Antoine Henrot. On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set. Mathematika, 2010, 56 (2), pp.245--265. ⟨10.1112/S0025579310000495⟩. ⟨hal-00443862⟩