Characterizations of ellipsoids by section-centroid location
The authors prove a generalization of a result of Blaschke. Let K be a convex body in \({\mathbb{R}}^ n\) with the following property: for any (n-1)- dimensional subspace M of \({\mathbb{R}}^ n\), the centroids of the sections \((x+M)\cap K\) are collinear (as x passes over all vectors x such that \((x+M)\cap int K\neq \emptyset)\); then (and only then) K is an ellipsoid. An immediate consequence is a positive answer to a problem raised by Gruber and Höbinger. Let \(k\in \{1,...,n-1\}\). The convex body K in \({\mathbb{R}}^ n\) is an ellipsoid if and only if for every k-dimensional subspace M of \(R^ n\), there exists an (n-k)-dimensional affine subspace L of \({\mathbb{R}}^ n\) so that all the centroids of k-dimensional sections of K parallel to M lie in L.
scientific article; zbMATH DE number 4119091