Orlicz Version of Mixed Mean Dual Affine Quermassintegrals

Abstract

In this paper, our main aim is to generalize the mixed mean dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new geometric operator by calculating the first Orlicz variation of the mixed mean dual affine quermassintegrals and call it the Orlicz mixed mean dual affine quermassintegrals. The fundamental notions and conclusions of the mixed mean dual affine quermassintegrals, and the Minkowski and Brunn–Minkowski inequalities for the mixed mean dual affine quermassintegrals are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual quermassintegrals are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the Lp-dual Minkowski inequality and Brunn–Minkowski inequality for the mixed mean dual affine quermassintegrals.