发表机构
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences; ELTE, Mathematical Institute; Georgia Institute of Technology; New York University（匈牙利科学院阿尔弗雷德·雷尼数学研究所; 罗兰大学数学研究所; 佐治亚理工学院; 纽约大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在群对称框架下研究加权对偶闵可夫斯基问题，推广了原点对称情形的存在性结果，给出了不同q取值下解存在的充要或充分条件，并推广了Henk-Pollehn的集中性质。
AI 中文摘要
本文研究了由Huang、Lutwak、Yang和Zhang首次提出的关于$\mathbb R^n$中凸体的加权$q$阶对偶曲率测度的加权对偶闵可夫斯基问题。大部分结果是在$G$-不变设定下获得的，即凸体在$O(n)$的闭子群$G$下不变。他们推广了先前关于原点对称凸体的对偶闵可夫斯基问题解的存在性结果。证明了：(i) 对于$q\in(0,1]$，$G$-不变加权对偶曲率测度的完整刻画；(ii) 对于$q\in(1,n)$，$G$-不变加权对偶闵可夫斯基问题解存在的必要且充分条件；(iii) 对于$q>1$，$G$-不变加权对偶曲率测度的充分条件；(iv) Henk-Pollehn对偶曲率测度集中性质的推广。
英文摘要
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.
