General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

Abstract

For convex bodies K in $\mathbb {R}^n$ containing the origin in their interiors, the general dual volume and the general dual Orlicz curvature measure $\widetilde{C}_{G, \psi }(K, \cdot )$ were recently introduced for certain classes of functions G and $\psi$. We extend these concepts to more general functions G 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 are provided which are required to study a Minkowski-type problem for the dual Orlicz curvature measure. The Minkowski problem asks to characterize Borel measures $\mu$ on the unit sphere for which there is a convex body K in $\mathbb {R}^n$ containing the origin such that $\mu$ equals $\widetilde{C}_{G, \psi }(K, \cdot )$, up to a constant. A major step in the analysis concerns discrete measures $\mu$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. Under mild conditions on G and $\psi$, solutions are obtained for general measures by an approximation argument. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $\mu$ is discrete or even.