The Weighted Dual Minkowski Problem Under Group Symmetry

Abstract

The paper studies the weighted dual Minkowski problem for the weighted $q$th dual curvature measure of convex bodies in ${\mathbb{R}}^n$ first posed by Huang, Lutwak, Yang, and Zhang. Most of the results are in the G-invariant setting, i.e. when the convex bodies are invariant under a closed subgroup $G$ of $O(n)$. They generalize previous results on the existence of solutions to the dual Minkowski problem for origin-symmetric convex bodies. It is proved: (i) a full characterization of the G-invariant weighted dual curvature measure for q\in(0,1], (ii) a necessary and sufficient condition for the existence of solutions to the G-invariant weighted dual Minkowski problem for q\in(1,n), (iii) a sufficient condition for the G-invariant weighted dual curvature measure for q>1, and (iv) an extension of Henk-Pollehn's dual curvature measure concentration property.