General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem I

TL;DR

This paper introduces a broad generalization of volume concepts for star bodies within the Orlicz-Brunn-Minkowski framework, establishing new curvature measures, variational formulas, and solving a Minkowski problem for these generalized volumes.

Contribution

It develops a unified theory of general volumes and dual Orlicz curvature measures, solving a Minkowski problem and deriving new inequalities in the Orlicz-Brunn-Minkowski setting.

Findings

Introduced a new general volume encompassing various dual volumes.

Established variational formulas for Orlicz linear combinations.

Solved the Minkowski problem for the new dual Orlicz curvature measure.

Abstract

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.