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

TL;DR

This paper extends a centroid collinearity characterization of ellipsoids to unbounded convex sets and identifies which affine hyperspheres satisfy this property, revealing that only quadrics do under certain conditions.

Contribution

It generalizes the centroid collinearity condition to unbounded convex sets and characterizes affine hyperspheres that satisfy this property as quadrics.

Findings

Ellipsoids, paraboloids, and one sheet of hyperboloids satisfy the centroid collinearity property.

Among affine hyperspheres, only quadrics meet the extended centroid condition.

Additional assumptions can force convex hypersurfaces with this property to be quadrics.

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.