Unique determination of convex polytopes by non-central sections
Let \(n\in\mathbb{N}\) and \(n\geq 2\). A subset \(K\) of \(\mathbb{R}^n\) is called a convex body if \(K\) is bounded, closed, convex and has nonempty interior. Let \(L\), \(K_1\), \(K_2\) be convex bodies in \(\mathbb{R}^n\). Assume that for each hyperplane \(H\) supporting \(L\) the \((n- 1)\)-dimensional volumes of the sections \(H\cap K_1\) and \(H\cap K_2\) coincide. In [Discrete Math. 241, No. 1--3, 79--96 (2001; Zbl 0999.52002), Conjecture 2] \textit{J. A. Barker} and \textit{D. G. Larman} conjectured that under the given assumptions one has \(K_1= K_2\). The author notices that if \(L\) is replaced by a single point, the question about the equality of \(K_1\) and \(K_2\) is well-known; this question can be answered by the analysis of injectivity properties of the spherical Radon transform. In the paper under review, the author confirms the conjecture of Barker and Larman for the case that \(L\) is a Euclidean ball and \(K_1\), \(K_2\) are \(n\)-dimensional polytopes. In the proof the cases \(n= 2\) and \(n\geq 3\) are considered separately, where the case \(n\geq 3\) is the main part of the proof. The author points out that the case \(n= 2\) is also covered by the result of \textit{G. Xiong}, \textit{Y. Ma} and \textit{W. Cheung}; see [J. Shanghai Univ. 12, No. 3, 200--203 (2008; Zbl 1174.52306)].
scientific article; zbMATH DE number 5851396