Fine structure of convex sets from asymmetric viewpoint
Given a convex body \({K}\) in the Euclidean space, the author defines a sequence of measures of the symmetry of \(K\) with respect to a fixed interior point \(O \in {\text{Int}}(K)\). Although it is not the original definition the author starts with, all terms of the above sequence are subsequently shown to be obtained from an affine invariant function \(\sigma\) defined on certain slices of \(K\). More precisely, \(\sigma_m(K)=\inf_{O\in F,\;{\text{dim}} F=m} \sigma (K \cap F)\), where \(F\) denotes an \(m\)-dimensional affine space passing through \(O\). The paper contains several results relating the sequence \(\sigma_m\) to the geometric properties of the body \(K\). In particular, the symmetry of \(K\) with respect to \(O\) implies that the singular set of the function \(\sigma\) consists of all the interior points of \(K\). It is shown, via constructive examples, that the converse does not hold. In general, to calculate the singular set of \(\sigma\) is difficult, hence the author provides a number of results which facilitate this task.
scientific article; zbMATH DE number 5888457