A quick proof of Höbinger-Burton-Larman's theorem
A slab in \(E^n\) in direction \(u\in S^{n-1}\) is a set \[ S=\bigl\{x\in E^n \mid \delta \leq \langle x,y \rangle\leq \alpha \bigr\} \] for some \(\alpha\) and \(\delta\). The following result, proved originally by Burton and Larman, is a solution of the Höbinger problem. Theorem (HBL). Let \(K\subset E^n\), \(n\geq 3\), be a convex body containing the origin and let \(S_1, \dots, S_{n-1}\) be slabs, at least one of which is not degenerate, in linearly independent directions \(u_1, \dots, u_{n-1}\), such that \(K\cap S_i= \emptyset\) for every \(i\). If for every point \(p\in S_i\) the projection of \(K\) from \(p\) into \(H_{u_i}\) is centrally symmetric, \(1\leq i\leq n-1\), then \(K\) is an ellipsoid. The authors present a new proof of the HBL Theorem, based on the notions of projective center and harmonic hyperplane of a convex body in \(E^n\). Definition reads as follows. Let \(P^n\) be the projective extension of \(E^n\) of \(K\) a convex body in \(E^n\). A point \(p\) and a hyperplane \(H\) are, respectively, a projective center of \(K\) and a harmonic hyperplane of \(K\) rel. to \(p\) provided that there exists a projective isomorphism \(f:P^n\to P^n\) such that \(f(K)\) is a centrally symmetric convex body in \(E^n\) with center \(f(p)\) and \(f(H)\) is the hyperplane at infinity.
scientific article; zbMATH DE number 955331