Revisiting floating bodies (Q346263)

Archimedes (\(\sim\) 287--212 B.C.) studied floating geometrical bodies. In the course of the following 2000 years, numerous results concerning this topic were achieved. The paper under review starts with a brief chronological exposition of these results. Aim of the article is a unification and generalization of the results on floating bodies in a multidimensional context. Thus, the authors provide characterizations for special quadrics in the real \(n\)-dimensional Euclidean space, for example they show:NEWLINENEWLINE``Let \(n\geq 1\). Let \(p_1,\dots, p_n>0\), and \(k>0\) be fixed, and \(p({\mathbf x})=\sum_{i=1}^{n}p_i^2x_i^2\) for all \({\mathbf x}\in \mathbb{R}^n\). Every hyperplane tangent to the paraboloid \(z=p({\mathbf x})+k^2\) cuts off from the paraboloid \(z=p({\mathbf x})\) an \((n+1)\)-dimensional compact set \({\mathcal S}\) of constant Euclidean volume equal to \(F(k^2)=p_1\dots p_nF_0(k^2)\).NEWLINENEWLINEConversely, let \(f\in C^{(2)}(\mathbb{R}^n)\) be real-valued function such that \(f({\mathbf x})>p({\mathbf x})\) for all \({\mathbf x}\in \mathbb{R}^n\). If every hyperplane tangent to the \(C^{(2)}\)-surface \(z=f({\mathbf x})\) cuts off from the paraboloid \(z=p({\mathbf x})\) a compact set of constant \((n+1)\)-dimensional Euclidean volume \(V\), then \(f({\mathbf x})=p({\mathbf x})+k^2\), where \(k^2=F^{-1}(V)\).''NEWLINENEWLINEAnalogous theorems are proved for hyperbolic surfaces and cones as well as elliptic surfaces.NEWLINENEWLINENine helpful figures accompany the considerations and computations.