Unique determination of convex polytopes by non-central sections
type: Item
instance of: scholarly article
title: Unique determination of convex polytopes by non-central sections
zbMATH Open document ID: 1220.52002
author: V. Yaskin
published in: Mathematische Annalen
publication date: 2011-02-14T00:00:00Z
review text: 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)].
reviewed by: Gennadiy Averkov
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 5851396
zbMATH Keywords: convex body
zbMATH Keywords: geometric tomography
zbMATH Keywords: hyperplane
zbMATH Keywords: polytope
zbMATH Keywords: section
MaRDI profile type: Publication
full work available at URL: s00208-010-0538-y
OpenAlex ID: W1976319098
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DOI: 10.1007/S00208-010-0538-Y
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description: scientific article; zbMATH DE number 5851396