Detecting symmetry in star bodies
type: Item
instance of: scholarly article
title: Detecting symmetry in star bodies
zbMATH Open document ID: 1261.52005
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-08-30T00:00:00Z
reviewed by: Erhard Heil
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 6074401
zbMATH Keywords: convex bodies
zbMATH Keywords: star bodies
zbMATH Keywords: origin-symmetry
zbMATH Keywords: cones
zbMATH Keywords: Fourier analysis
zbMATH Keywords: tomography
author: D. Ryabogin
author: V. Yaskin
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.05.022
OpenAlex ID: W2118580292
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DOI: 10.1016/J.JMAA.2012.05.022
review text: It was shown by \textit{E. Makai, jun.}, \textit{H. Martini} and \textit{T. Ódor} [Mathematika 47, No. 1--2, 19--30 (2000; Zbl 1012.52008)] that a convex body \(K\) in \(\mathbb{R}^n\) is origin-symmetric if all \((n-1)\)-volume functions of parallel hyperplane sections have a critical value at the origin.NEWLINENEWLINE In the present paper \(K\) may be a star body and the hyperplanes are replaced by circular hypercones with centers at the origin. It is shown that \(K\) is origin-symmetric if all the \((n-1)\)-volume functions of intersections with \(K\) have critical values when the cone degenerates to a hyperplane. The proof uses Fourier transforms of distributions as described by \textit{A. Koldobsky} [Fourier analysis in convex geometry. Mathematical Surveys and Monographs 116. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1082.52002)]. The paper also gives a new proof of the theorem of Makai, Martini and Ódor [loc. cit.] in the case of star bodies.
description: scientific article; zbMATH DE number 6074401