On the structure of convex sets with symmetries

Abstract

Asymmetry of a compact convex body \({\mathcal L \subset {\bf R}^n}\) viewed from an interior point \({\mathcal O}\) can be measured by considering how far \({\mathcal L}\) is from its inscribed simplices that contain \({\mathcal O}\). This leads to a measure of symmetry \({\sigma(\mathcal L, \mathcal O)}\). The interior of \({\mathcal L}\) naturally splits into regular and singular sets, where the singular set consists of points \({\mathcal O}\) with largest possible \({\sigma(\mathcal L, \mathcal O)}\). 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 \({\mathcal O}\), \({\sigma(\mathcal L, .)}\) is concave on the regular set. It is natural to ask to what extent does concavity of \({\sigma(\mathcal L, .)}\) extend to the whole (interior) of \({\mathcal L}\). It has been shown earlier that in dimension two, \({\sigma(\mathcal L, .)}\) is concave on \({\mathcal L}\). In this paper, we show that in dimensions greater than two, for a centrally symmetric compact convex body \({\mathcal L}\), \({\sigma(\mathcal L, .)}\) is a non-concave function provided that \({\mathcal L}\) 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.