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

For convex bodies K in R-n containing the origin in their interiors, the general dual volume and the general dual Orlicz curvature measure (C) over tilde (G,psi) (K, center dot) 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 Rn containing the origin such that mu equals (C) over tilde (G,psi) (K, center dot), 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.