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

Junjie Shan

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 ($p = 0$).

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

Abstract

In this paper, we study the dual Minkowski problem for all 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 Minkowski problem (), the Aleksandrov problem (), and the dual Minkowski problem ().

Paper Structure

This paper contains 7 sections, 36 theorems, 146 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classification of Group-Invariant Convex Bodies
  4. Geometric Characterization of Irreducible Closed Subgroups
  5. Existence of Solutions to the $L_{p}$ Dual Minkowski Problem for $q\neq 0$
  6. The $L_{p}$ Aleksandrov Problem
  7. Extension to Non-Closed Subgroups

Key Result

Theorem 1.1

Let $q\in \mathbb{R}$, $p \in \mathbb{R}$, $G$ be an irreducible subgroup of $\mathrm{O}(n)$ with $n \geq 2$, and $Q$ be a $G$-invariant star body in $\mathbb{R}^n$. For any non-zero finite Borel measure $\mu$ on $S^{n-1}$, $\mu$ is $G$-invariant if and only if there exists a $G$-invariant convex bo and

Figures (1)

  • Figure :

Theorems & Definitions (67)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1: lyz2018
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 57 more