We consider a family of functionals $J$ to be maximized over the planarconvex sets $K$ for which the perimeter and Steiner point have been fixed.Assuming that $J$ is the integral of a quadratic expression in the supportfunction $h$, we show that the maximizer is always either a triangle or a linesegment (which can be considered as a collapsed triangle). Among the concreteconsequences 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 inthe sense of the $L^2$ distance is always a line segment. We also prove thesame property for the Hausdorff distance.
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机译：我们认为一族函数$ J $在固定了周长和Steiner点的平面凸集$ K $上是最大化的，假设$ J $是支持函数$ h $中二次表达式的积分，表明最大化器始终是三角形或线段（可以视为折叠三角形）。在主要定理的具体结果中，有一个事实是，给定有限周长的任何凸体$ K_1 $，我们认为该类中的集合在$ L ^ 2 $距离意义上最远就是线段。我们还证明了Hausdorff距离的相同属性。
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