The Log-Aleksandrov–Fenchel Inequality

Abstract

In this paper, we first propose and establish a new Aleksandrov–Fenchel type inequality involving logarithms by introducing two new concepts of mixed volume measure and $$L_{p}$$-multiple mixed volume measure, and using the $$L_{p}$$-Aleksandrov–Fenchel inequality for the $$L_{p}$$-multiple mixed volumes. The new Aleksandrov–Fenchel inequality involving logarithms in special case yields the classical Aleksandrov–Fenchel inequality, and four recent logarithmic Minkowski inequalities, which are logarithmic Minkowski inequality for mixed volumes, $$L_{p}$$-mixed volumes, quermassintegrals and p-mixed quermassintegrals, respectively. Moreover, the logarithmic Aleksandrov–Fenchel inequality is also derived.