Uniqueness and Continuity of the Solution to $$L_p$$ Dual Minkowski Problem

Abstract

Lutwak et al. (Adv Math 329:85–132, 2018) introduced the \(L_p\) dual curvature measure that unifies several other geometric measures in dual Brunn–Minkowski theory and Brunn–Minkowski theory. Motivated by works in Lutwak et al. (Adv Math 329:85–132, 2018), we consider the uniqueness and continuity of the solution to the \(L_p\) dual Minkowski problem. To extend the important work (Theorem A) of LYZ to the case for general convex bodies, we establish some new Minkowski-type inequalities which are closely related to the optimization problem associated with the \(L_p\) dual Minkowski problem. When \(q< p\), the uniqueness of the solution to the \(L_p\) dual Minkowski problem for general convex bodies is obtained. Moreover, we obtain the continuity of the solution to the \(L_p\) dual Minkowski problem for convex bodies.