The general dual-polar Orlicz-Minkowski problem

2019/10/05 by Sudan Xing, Xing, Sudan, Deping Ye +3
Mathematics · #52A20 #52A38 #52A39 #52A40 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1910.02178

openalex publication_date 2019/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper gives a systematic study to the general dual-polar Orlicz-Minkowski problem (e.g., Problem \refgeneral-dual-polar). This problem involves the general dual volume \widetildeV_G(⋅) recently proposed in \citeGHWXY, GHXY in order to study the general dual Orlicz-Minkowski problem. As \widetildeV_G(⋅) 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 \citeGHWXY, GHXY and "dual" to the newly proposed polar Orlicz-Minkowski problem in \citeLuoYeZhu. 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.

Citations

• Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

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