Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

Abstract

Our main aim is to generalize the classical mixed volumeV(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 itOrlicz multiple mixed volumeof convex bodiesK1,…,Kn, andLn, denoted byVφ(K1,…,Kn,Ln), which involves(n+1)convex bodies inRn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities ofLp-multiple mixed volumeVp(K1,…,Kn,Ln)are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yieldsLp-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.

The Dual Orlicz–Aleksandrov–Fenchel Inequality

“…The same concepts and inequalities are derived by Xi, Jin and Leng [27] using a new geometric symmetry technique. Other articles on this theory can be found in the literature [28][29][30][31][32][33][34][35].…”

The dual Orlicz-Aleksandrov-Fenchel inequality

“…The Orlicz centroid inequality for star bodies was introduced in [61] which is an extension from convex to star bodies. The other articles advance the theory can be found in literatures [19], [25], [27], [45] and [55].…”