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

TL;DR

This paper investigates the $L_p$ dual Minkowski problem for group-invariant convex bodies across all real parameters, unifying several classical problems and establishing solution existence from an algebraic perspective.

Contribution

It introduces a unified algebraic approach to prove the existence of solutions for the $L_p$ dual Minkowski problem for all $q, p \, \in \mathbb{R}$, including key special cases.

Findings

Established existence of solutions for group-invariant convex bodies.

Unified treatment of $L_p$ Minkowski, Aleksandrov, and dual Minkowski problems.

Covered cases not necessarily symmetric about the origin.

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 ($p = 0$).