Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes. (16th September 2018) - Zhao, Chang-Jian
Abstract : Our main aim is to generalize the classical mixed volumeV ( 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 bodiesK 1, …, K n, andL n, denoted byV φ ( K 1, …, K n, L n ), which involves( n + 1 ) convex bodies inR 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 ofL p -multiple mixed volumeV p ( K 1, …, K n, L n ) are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yieldsL 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.