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 4.1). This problem involves the general dual volume \({\widetilde{V}}_G(\cdot )\) recently proposed in Gardner et al. (Calc Var PDE 58:35, 2019; 59:33, 2020) in order to study the general dual Orlicz–Minkowski problem. As \({\widetilde{V}}_G(\cdot )\) extends the volume and the qth dual volume, the general dual-polar Orlicz–Minkowski problem is “polar" to the recently initiated general dual Orlicz–Minkowski problem in Gardner et al. (Calc Var PDE 58:35, 2019; 59:33, 2020) and “dual" to the newly proposed polar Orlicz–Minkowski problem in Luo et al. (Indiana Univ Math J 69:385–420, 2020). The existence, uniqueness and continuity, 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.