  Lutwak, Yang and Zhang \cite{LYZ2018} 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 \cite{LYZ2018}, we
consider the uniqueness and continuity of the solution to the $L_p$ dual
Minkowski problem. To extend the important work (Theorem \ref{uniquepolytope})
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.
