An Affine Invariant Minkowski Problem

In Euclidean space, the generalised Minkowski problem asks, for a given finite Radon measure $\mu$ on the unit sphere ${\mathbb{S}}^d$, to find a compact convex set $K$ with area measure $\mu$. For convex sets in the Minkowski space invariant under an affine deformation of a uniform lattice of ${\mathrm{S}\mathrm{O}}_0(d,1)$, 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 $\mathrm{S}\mathrm{L}({\mathbb{R}}^{d+1})$ 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.