A Characterization of Quadrics Among Affine Hyperspheres by Section-Centroid Location

Abstract

A theorem of Meyer and Reisner characterizes ellipsoids by the collinearity of centroids of parallel sections: if $\Omega\subset\mathbb{R}^{n+1}$ is a convex body such that for every $n$-dimensional subspace $M\subset\mathbb{R}^{n+1}$ the centroids of the sections $(x+M)\cap \Omega$ are collinear, then $\Omega$ is an ellipsoid. We study natural extensions of this centroid-collinearity condition to unbounded convex sets. In particular, we show that among affine hyperspheres, precisely the ellipsoids, paraboloids and one sheet of a two-sheeted hyperboloid satisfy this property. We also identify additional assumptions under which any convex hypersurface with this property must necessarily be a quadric.