Dual Orlicz-Brunn-Minkowski theory: Orlicz $\varphi$-radial addition, Orlicz $L_{\phi}$-dual mixed volume and related inequalities

TL;DR

This paper introduces a new dual Orlicz-Brunn-Minkowski framework for star bodies, establishing inequalities and formulas for dual mixed volumes and surface areas, advancing the geometric analysis in convex geometry.

Contribution

It develops the dual Orlicz-Brunn-Minkowski theory, including new addition operations and inequalities for star bodies, extending classical convex geometric concepts.

Findings

Established dual Orlicz-Brunn-Minkowski inequality.

Derived a formula for Orlicz $L_{\phi}$-dual mixed volume.

Proved dual Orlicz-Minkowski, isoperimetric, and Urysohn inequalities.

Abstract

This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz $\varphi$-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz $\varphi$-radial addition of two star bodies, we derive a formula for the Orlicz $L_{\phi}$-dual mixed volume. Moreover, a dual Orlicz-Minkowski inequality for the Orlicz $L_{\phi}$-dual mixed volume, a dual Orlicz isoperimetric inequality for the Orlicz $L_{\phi}$-dual surface area and a dual Orlicz-Urysohn inequality for the Orlicz $L_{\phi}$-harmonic mean radius are proved.