The Dual Minkowski Problem under Group Actions

TL;DR

This paper investigates the dual Minkowski problem with group symmetry, providing a complete existence characterization for G-invariant convex bodies across different parameter ranges.

Contribution

It offers a comprehensive solution to the dual Minkowski problem under group actions, extending known results to the G-invariant setting and connecting to the logarithmic Minkowski problem.

Findings

Characterization of existence conditions for G-invariant convex bodies

Recovery of the origin-symmetric case when G={±I}

Conditions based on measure concentration on G-invariant subspaces

Abstract

In this paper, we study the dual Minkowski problem under group symmetry. For $0<q\le n$, we give a complete existence characterization in the framework of $G$-invariant convex bodies, recovering the origin-symmetric setting when $G=\{\pm I\}$. The necessary and sufficient conditions concern the concentration of the measure on $G$-invariant subspaces, both in the range $0<q<n$ and at the critical endpoint $q=n$, where the problem becomes the logarithmic Minkowski problem.