An Affine Invariant Minkowski Problem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918484891402240 |
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| author | Ablondi, Antoine |
| author_facet | Ablondi, Antoine |
| contents | In Euclidean space, the generalised Minkowski problem asks, for a given finite Radon measure $μ$ on the unit sphere $\mathbb{S}^d$, to find a compact convex set $K$ with area measure $μ$. For convex sets in the Minkowski space invariant under an affine deformation of a uniform lattice of $\mathrm{SO}_0(d,1)$, the analogous Minkowski problem was considered and solved by Barbot--Béguin--Zeghib (partially) and Bonsante--Fillastre. By a theorem of Mess--Barbot--Bonsante, that also solves the Minkowski problem in flat Lorentzian spacetimes with compact hyperbolic Cauchy surface.
We consider convex domains of the oriented real affine space $\mathbb{R}^{d+1}$ which are invariant under a subgroup of affine transformations obtained by adding translation parts to a discrete subgroup of $\mathrm{SL} (\mathbb{R}^{d+1})$ dividing a convex cone. We prove that those convex domains satisfy a local Steiner Formula, allowing to introduce natural area measures and define an affine invariant Minkowski problem. We then solve that Minkowski problem through a variational method using the convexity of a covolume functional. We also give an interpretation of those results in some "affine spacetimes", which were introduced by the author in a preceding work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03791 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Affine Invariant Minkowski Problem Ablondi, Antoine Differential Geometry In Euclidean space, the generalised Minkowski problem asks, for a given finite Radon measure $μ$ on the unit sphere $\mathbb{S}^d$, to find a compact convex set $K$ with area measure $μ$. For convex sets in the Minkowski space invariant under an affine deformation of a uniform lattice of $\mathrm{SO}_0(d,1)$, the analogous Minkowski problem was considered and solved by Barbot--Béguin--Zeghib (partially) and Bonsante--Fillastre. By a theorem of Mess--Barbot--Bonsante, that also solves the Minkowski problem in flat Lorentzian spacetimes with compact hyperbolic Cauchy surface. We consider convex domains of the oriented real affine space $\mathbb{R}^{d+1}$ which are invariant under a subgroup of affine transformations obtained by adding translation parts to a discrete subgroup of $\mathrm{SL} (\mathbb{R}^{d+1})$ dividing a convex cone. We prove that those convex domains satisfy a local Steiner Formula, allowing to introduce natural area measures and define an affine invariant Minkowski problem. We then solve that Minkowski problem through a variational method using the convexity of a covolume functional. We also give an interpretation of those results in some "affine spacetimes", which were introduced by the author in a preceding work. |
| title | An Affine Invariant Minkowski Problem |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.03791 |