The general dual-polar Orlicz-Minkowski problem

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.