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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.06107 |
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| _version_ | 1866915719671709696 |
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| author | Borentain, Alexandre |
| author_facet | Borentain, Alexandre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06107 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Characterization of Quadrics Among Affine Hyperspheres by Section-Centroid Location Borentain, Alexandre Differential Geometry 52A20, 53A15 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. |
| title | A Characterization of Quadrics Among Affine Hyperspheres by Section-Centroid Location |
| topic | Differential Geometry 52A20, 53A15 |
| url | https://arxiv.org/abs/2601.06107 |