The general dual-polar Orlicz-Minkowski problem

TL;DR

This paper systematically studies the general dual-polar Orlicz-Minkowski problem, establishing existence, uniqueness, and continuity of solutions, and explores polytopal solutions and variations related to Orlicz-Petty bodies.

Contribution

It introduces a comprehensive framework for the general dual-polar Orlicz-Minkowski problem, extending previous concepts and providing new existence, uniqueness, and solution characterizations.

Findings

Established existence, continuity, and uniqueness of solutions.

Provided polytopal solutions and counterexamples for discrete measures.

Discussed variations leading to the general Orlicz-Petty bodies.

Abstract

This paper gives a systematic study to the general dual-polar Orlicz-Minkowski problem (e.g., Problem \ref{general-dual-polar}). This problem involves the general dual volume $\widetilde{V}_G(\cdot)$ recently proposed in \cite{GHWXY, GHXY} in order to study the general dual Orlicz-Minkowski problem. As $\widetilde{V}_G(\cdot)$ extends the volume and the $q$th dual volume, the general dual-polar Orlicz-Minkowski problem is "polar" to the recently initiated general dual Orlicz-Minkowski problem in \cite{GHWXY, GHXY} and "dual" to the newly proposed polar Orlicz-Minkowski problem in \cite{LuoYeZhu}. The existence, continuity and uniqueness, if applicable, for the solutions to the general dual-polar Orlicz-Minkowski problem are established. Polytopal solutions and/or counterexamples to the general dual-polar Orlicz-Minkowski problem for discrete measures are also provided. Several variations of the general dual-polar Orlicz-Minkowski problem are discussed as well, in particular the one leading to the general Orlicz-Petty bodies.