A flow approach to the Musielak-Orlicz-Gauss image problem

TL;DR

This paper introduces a flow-based method to solve the extended Musielak-Orlicz-Gauss image problem, generalizing previous solutions and addressing many previously unsolved cases in convex geometric analysis.

Contribution

It develops a novel parabolic flow approach involving Musielak-Orlicz functions to solve the extended Musielak-Orlicz-Gauss image problem, broadening the scope of existing Minkowski and Gauss image solutions.

Findings

Provides solutions to the extended Musielak-Orlicz-Gauss image problem.

Generalizes many known Minkowski and Gauss image problems.

Addresses previously unsolved cases in convex geometric analysis.

Abstract

In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,\Psi,\lambda}(\Omega,\cdot)$ of convex body $\Omega$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.