Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_{\varphi }$ affine and geominimal surface areas

This paper aims to develop basic theory for the dual Orlicz $L_{\varphi }$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequality, cyclic inequality, Santal'{o} style inequality and Alexander-Fenchel type inequality are established.