The affine log-Aleksandrov–Fenchel inequality

Abstract

We establish a new affine log-Aleksandrov-Fenchel inequality for mixed affine quermassintegrals by introducing new concepts of affine and multiple affine measures, and using the newly established Aleksandrov-Fenchel inequality for multiple mixed affine quermassintegrals. Our new inequality yields as special cases the classical Aleksandrov-Fenchel inequality and the Lp-affine log-Aleksandrov-Fenchel inequality. The affine log-Minkowski and log-Aleksandrov-Fenchel inequalities are also derived.