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

Alexandre Borentain

TL;DR

This work extends the section-centroid collinearity property (SCCP) to unbounded convex sets and classifies which affine hyperspheres satisfy SCCP, showing that the ellipsoid, paraboloid, and one sheet of a two-sheeted hyperboloid are exactly those with SCCP among affine hyperspheres. It develops a cut-volume framework and adapts Meyer–Reisner lemmas to unbounded settings, introducing a $C^1$ cut-volume functional with a centroid formula $x(a)=\nabla V(a)/\langle a,\nabla V(a)\rangle$ and showing constancy of $V(a)$ under suitable hypotheses. The paper then connects centroid-line behavior to recession cones, proving that centroid directions for $\Omega$ and its recession cone $\mathcal C$ are parallel, and in the asymptotic-to-cone case, the boundary is a quadric; this ties centroid geometry to equi-affine surface theory via Kim’s quadric characterizations. Remaining open questions concern whether SCCP forces affine-hypersphere structure in general and how cases with intermediate recession-dimension cones influence the quadric classification, highlighting a rich interplay between centroid geometry, asymptotics, and quadric classification in affine differential geometry.

Abstract

A theorem of Meyer and Reisner characterizes ellipsoids by the collinearity of centroids of parallel sections: if $Ω\subset\mathbb{R}^{n+1}$ is a convex body such that for every $n$-dimensional subspace $M\subset\mathbb{R}^{n+1}$ the centroids of the sections $(x+M)\cap Ω$ are collinear, then $Ω$ 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.

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

TL;DR

This work extends the section-centroid collinearity property (SCCP) to unbounded convex sets and classifies which affine hyperspheres satisfy SCCP, showing that the ellipsoid, paraboloid, and one sheet of a two-sheeted hyperboloid are exactly those with SCCP among affine hyperspheres. It develops a cut-volume framework and adapts Meyer–Reisner lemmas to unbounded settings, introducing a cut-volume functional with a centroid formula and showing constancy of under suitable hypotheses. The paper then connects centroid-line behavior to recession cones, proving that centroid directions for and its recession cone are parallel, and in the asymptotic-to-cone case, the boundary is a quadric; this ties centroid geometry to equi-affine surface theory via Kim’s quadric characterizations. Remaining open questions concern whether SCCP forces affine-hypersphere structure in general and how cases with intermediate recession-dimension cones influence the quadric classification, highlighting a rich interplay between centroid geometry, asymptotics, and quadric classification in affine differential geometry.

Abstract

A theorem of Meyer and Reisner characterizes ellipsoids by the collinearity of centroids of parallel sections: if is a convex body such that for every -dimensional subspace the centroids of the sections are collinear, then 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.
Paper Structure (14 sections, 6 theorems, 59 equations, 1 figure)

This paper contains 14 sections, 6 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Organization.
  3. Notation and basic objects
  4. Convergence to cones
  5. Gauss map and centroid loci
  6. Core lemmas and classification results
  7. Differentiability of the cut-volume functional
  8. Floating body characterization
  9. Asymptotic behavior of centroid lines and recession cones
  10. Conclusion and open problems
  11. Case 1: full-dimensional recession cone, $\Omega$ asymptotic to its cone.
  12. Case 2: one-dimensional recession cone.
  13. Case 3: full-dimensional recession cone, not asymptotic to the cone.
  14. Case 4: intermediate-dimensional recession cone.

Key Result

Theorem 1

Let $\Omega\subset\mathbb{R}^{n+1}$ be a strictly convex domain with smooth boundary $\partial\Omega$, equipped with its Blaschke (equi-affine) normalization, so that $\partial\Omega$ is an affine hypersphere. Assume that for every $n$--dimensional linear subspace $M\subset\mathbb{R}^{n+1}$ such tha

Figures (1)

  • Figure 1: The function $f(x)=e^x$.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Definition 1: Hyperplanes, sections, and centroids
  • Definition 2: Minkowski functional and support function
  • Definition 3: Recession cone
  • Definition 4: Blow-down convergence
  • Definition 5: Asymptotic convergence
  • Definition 6: Centroid curve and centroid line
  • Definition 7: Cut--volume functional
  • Lemma 4
  • ...and 8 more