General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem II
The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $ψ$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $μ$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $μ$ equals $\deV(K, \cdot)$, up to a constant. A major step in the analysis concerns discrete measures $μ$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures $μ$, we use an approximation argument. This approach is also applied to the case where $G$ is decreasing and $ψ$ is increasing, and hence augments our previous work. When the measures $μ$ are even, solutions that are origin-symmetric convex bodies are also provided under some mild conditions on $G$ and $ψ$. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $μ$ is discrete or even.