An Affine Invariant Minkowski Problem

TL;DR

This paper extends the Minkowski problem to affine-invariant convex domains in real affine space, introducing new area measures and solving it via a variational approach, with applications to affine spacetimes.

Contribution

It formulates and solves an affine invariant Minkowski problem for convex domains invariant under certain affine transformations, expanding classical Minkowski theory.

Findings

Established a local Steiner Formula for these convex domains.

Introduced natural area measures for the affine-invariant setting.

Solved the Minkowski problem using a variational method with a covolume functional.

Abstract

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 $\mathrm{SO}_0(d,1)$, the analogous Minkowski problem was considered and solved by Barbot--B\'eguin--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.