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The $L_p$ chord Minkowski problem for super-critical exponent

Shibing Chen, Qi-Rui Li, Yuanyuan Li

TL;DR

The paper addresses the super-critical $L_p$ chord Minkowski problem by employing a nonlocal Gauss curvature flow that is the gradient flow of a carefully crafted functional $\\mathcal{J}$. A simplified topological argument (via a Brouwer fixed-point approach) provides the needed initial data, and a priori $C^2$-estimates yield long-time smooth evolution. The authors prove existence of uniformly convex, $C^{3,\,\alpha}$ solutions for density $f$ with bounds, and extend to $L^{\infty}$ densities through an approximation argument, establishing convergence to a solution of the governing Monge–Ampère type equation. This work extends the scope of the $L_p$ chord Minkowski problem to the super-critical regime and highlights a streamlined topology-based component alongside a nonlocal flow analysis.

Abstract

The $L_p$ chord Minkowski problem was recently introduced by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the $L_p$ chord measure of a convex body. In this paper, we solve the $L_p$ chord Minkowski problem for the super-critical exponents by combining a nonlocal Gauss curvature flow introduced in \cite{HHLW exi} and a topological argument developed in \cite{GLW2022}. Notably, we provide a simplified argument for the topological part.

The $L_p$ chord Minkowski problem for super-critical exponent

TL;DR

The paper addresses the super-critical chord Minkowski problem by employing a nonlocal Gauss curvature flow that is the gradient flow of a carefully crafted functional . A simplified topological argument (via a Brouwer fixed-point approach) provides the needed initial data, and a priori -estimates yield long-time smooth evolution. The authors prove existence of uniformly convex, solutions for density with bounds, and extend to densities through an approximation argument, establishing convergence to a solution of the governing Monge–Ampère type equation. This work extends the scope of the chord Minkowski problem to the super-critical regime and highlights a streamlined topology-based component alongside a nonlocal flow analysis.

Abstract

The chord Minkowski problem was recently introduced by Lutwak, Xi, Yang and Zhang, which seeks to determine the necessary and sufficient conditions for a given finite Borel measure such that it is the chord measure of a convex body. In this paper, we solve the chord Minkowski problem for the super-critical exponents by combining a nonlocal Gauss curvature flow introduced in \cite{HHLW exi} and a topological argument developed in \cite{GLW2022}. Notably, we provide a simplified argument for the topological part.

Paper Structure

This paper contains 5 sections, 10 theorems, 127 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. A priori estimates for solutions to the Gauss curvature flow
  4. Properties of the functional and the initial condition
  5. Proof of Theorem \ref{['A']} and Theorem \ref{['B']}

Key Result

Theorem 1.1

Let $p<-n-q+1$, $3< q<n+1$, and $\mu$ be a finite Borel measure on $\mathbb{S}^{n-1}$ with density $f$. If $f\in C^{1,1}(\mathbb{S}^{n-1})$ and $\frac{1}{\Lambda}<f<\Lambda$ for some constant $\Lambda>0$, then there exists a uniformly convex, positive, $C^{3,\alpha}$ solution to MAeq, where $\alpha

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Theorem 5.5 in LXZY2022
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['C2']}
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • ...and 7 more