Abstract: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.
| Comments: | 34 pages, 7 figures |
| Subjects: | Metric Geometry (math.MG); Differential Geometry (math.DG) |
| MSC classes: | Primary: 52A38 Secondary: 52A55, 53C20, 53C60 |
| Cite as: | arXiv:2605.25122 [math.MG] |
| (or arXiv:2605.25122v1 [math.MG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25122 arXiv-issued DOI via DataCite (pending registration) |
From: Florian Besau [view email]
[v1]
Sun, 24 May 2026 15:08:42 UTC (373 KB)
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