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

Abstract

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