The $L_p$ Dual Minkowski Problem for Group-Invariant Convex Bodies

In this paper, we study the $L_p$ dual Minkowski problem for all $q,p\in \mathbb{R}$ from an algebraic perspective. We establish the existence of solutions for group-invariant convex bodies (not necessarily origin-symmetric), thereby covering three fundamental problems as special cases: the $L_p$ Minkowski problem ($q=n$), the $L_p$ Aleksandrov problem ($q=0$), and the dual Minkowski problem ($p=0$).