Illumination Bodies in Projective Geometries
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914597855821824 |
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| author | Assouline, Rotem Besau, Florian Werner, Elisabeth M. |
| author_facet | Assouline, Rotem Besau, Florian Werner, Elisabeth M. |
| contents | We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings, generalizing the affine surface area in Euclidean space.
The proof is based on a general result on the derivative of weighted volumes of weighted illumination bodies in Euclidean space. In the appendix, we give some explicit examples for non-Euclidean illumination bodies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25122 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Illumination Bodies in Projective Geometries Assouline, Rotem Besau, Florian Werner, Elisabeth M. Metric Geometry Differential Geometry Primary: 52A38 Secondary: 52A55, 53C20, 53C60 We extend the notion of illumination bodies to Riemannian spaces of constant curvature and to projective Finsler geometries. We prove that the derivative of their volume defines a notion of surface area for convex bodies in these settings, generalizing the affine surface area in Euclidean space. The proof is based on a general result on the derivative of weighted volumes of weighted illumination bodies in Euclidean space. In the appendix, we give some explicit examples for non-Euclidean illumination bodies. |
| title | Illumination Bodies in Projective Geometries |
| topic | Metric Geometry Differential Geometry Primary: 52A38 Secondary: 52A55, 53C20, 53C60 |
| url | https://arxiv.org/abs/2605.25122 |