On the Lp dual Minkowski problem

The Lp dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the Lp Brunn-Minkowski theory and the dual Brunn-Minkowski theory. The characterization problem for Lp dual curvature measure, called the Lp dual Minkowski problem, is a fundamental problem in this unifying theory. The Lp dual Minkowski problem contains the Lp Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the Lp dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated.