Asymmetry of a compact convex body \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal L \subset {\bf R}^n}$$\end{document} viewed from an interior point \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal O}$$\end{document} can be measured by considering how far \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal L}$$\end{document} is from its inscribed simplices that contain \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal O}$$\end{document}. This leads to a measure of symmetry \documentclass[12pt]{minimal}
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\begin{document}$${\sigma(\mathcal L, \mathcal O)}$$\end{document}. The interior of \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal L}$$\end{document} naturally splits into regular and singular sets, where the singular set consists of points \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal O}$$\end{document} with largest possible \documentclass[12pt]{minimal}
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\begin{document}$${\sigma(\mathcal L, \mathcal O)}$$\end{document}. In general, to calculate the singular set of a compact convex body is difficult. In this paper we determine a large subset of the singular set in centrally symmetric compact convex bodies truncated by hyperplane cuts. As a function of the interior point \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal O}$$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$${\sigma(\mathcal L, .)}$$\end{document} is concave on the regular set. It is natural to ask to what extent does concavity of \documentclass[12pt]{minimal}
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\begin{document}$${\sigma(\mathcal L, .)}$$\end{document} extend to the whole (interior) of \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal L}$$\end{document}. It has been shown earlier that in dimension two, \documentclass[12pt]{minimal}
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\begin{document}$${\sigma(\mathcal L, .)}$$\end{document} is concave on \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal L}$$\end{document}. In this paper, we show that in dimensions greater than two, for a centrally symmetric compact convex body \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal L}$$\end{document}, \documentclass[12pt]{minimal}
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\begin{document}$${\sigma(\mathcal L, .)}$$\end{document} is a non-concave function provided that \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal L}$$\end{document} has a codimension one simplicial intersection. This is the case, for example, for the n-dimensional cube, n ≥ 3. This non-concavity result relies on the fact that a centrally symmetric compact convex body has no regular points.
