Orlicz-Aleksandrov-Fenchel inequality for Orlicz multiple mixed volumes (Q1624152)

Summary: Our main aim is to generalize the classical mixed volume \(V(K_1, \ldots, K_n)\) 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 \textit{Orlicz multiple mixed volume} of convex bodies \(K_1, \ldots, K_n\), and \(L_n\), denoted by \(V_\varphi(K_1, \ldots, K_n, L_n)\), which involves \((n + 1)\) convex bodies in \(\mathbb{R}^n\). 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 \(V_p(K_1, \ldots, K_n, L_n)\) 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.