The $L_{p}$ Dual Minkowski Problem for Group-Invariant Convex Bodies

Abstract

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 ($p = 0$).