Detecting symmetry in star bodies
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.
scientific article; zbMATH DE number 6074401
scientific article