  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.
