Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Abstract EN

Our main aim is to generalize the classical mixed volume V(K1,…,Kn) 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 K1,…,Kn, and Ln, denoted by Vφ(K1,…,Kn,Ln), which involves (n+1) convex bodies in Rn.

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 Lp-multiple mixed volume Vp(K1,…,Kn,Ln) are also derived.

The Orlicz-Aleksandrov-Fenchel inequality in special cases yields Lp-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.