On the Lp dual Minkowski problem

The (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. (C) 2018 Elsevier Inc. All rights reserved.