Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_ϕ$ affine and geominimal surface areas

Abstract

This paper aims to develop basic theory for the dual Orlicz $L_ϕ$ 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, Santaló style inequality and Alexander-Fenchel type inequality are established.

Some changes are made in this version. The Orlicz $L_ϕ$-dual mixed volume defined in arXiv:1407.7311 (instead of arXiv:1404.6991) is used to define the dual Orlicz $L_ϕ$ affine and geominimal surface areas. Moreover, the proofs are simplified