Inverse Gauss curvature flows and Orlicz Minkowski problem (Q2104146)

The Orlitz Minkowski problem is the following. Suppose \(\varphi\) is a positive continuous function defined in \((0, + \infty)\), and \(\mu\) is a Borel measure on the unit sphere \(\mathcal{S}^{n-1}\). What are the necessary and sufficient conditions for \(\mu\), so that there exists a convex body \(K\) and a constant \(\lambda > 0\) such that \[ \lambda \varphi(h) d S_K = d \mu, \] where \(h\) and \(S_K\) is the support function and the surface area measure of \(K\), respectively? \textit{C. Haberl} et al. [Adv. Math. 224, No. 6, 2485--2510 (2010; Zbl 1198.52003)] proved the existence of a solution in the special case that \(\mu\) is even and \(\varphi\) satisfies some smoothness conditions. The aim of the paper is to extend this result by dropping the condition that \(\mu\) is even. The main tool of the authors is an inverse Gauss curvature flow defined on convex hypersurfaces.