On the L_p dual Minkowski problem

The L_p dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the L_p Brunn–Minkowski theory and the dual Brunn–Minkowski theory. The characterization problem for L_p dual curvature measure, called the L_p dual Minkowski problem, is a fundamental problem in this unifying theory. The L_p dual Minkowski problem contains the L_p 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 L_p 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.