We prove that the largest convex shape that can be placed inside a given convex shape \(Q \subset \mathbb {R}^{d}\) in any desired orientation is the largest inscribed ball of Q. The statement is true both when “largest” means “largest volume” and when it means “largest surface area”. The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.

中文翻译：

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反挂屋问题

我们证明了可以放置在给定凸起形状内的最大凸状 \（Q \子集\ mathbb {R} ^ {d} \）以任何所需的取向是最大的内切球 Q。当“最大”表示“最大体积”和“最大表面积”时，该陈述都是正确的。球是唯一的解决方案，除非在二维情况下最大化周长。