Orlicz-Aleksandrov-Fenchel Inequality or Orlicz Multiple Mixed Volumes

Our main aim is to generalize the classical mixed volume V(K-1, . . . , K-n) and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call it Orlicz multiple mixed volume of convex bodies K-1, . . . , K-n, and L-n, denoted by V-phi( K-1, . . . , K-n, L-n), which involves (n + 1) convex bodies in R-n. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities of L-p-multiple mixed volume V-p(K-1, . . . , K-n, L-n) are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yields L-p-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.