On Aleksandrov-Fenchel Inequalities for <Emphasis Type="Italic">k</Emphasis>-Convex Domains

In this lecture notes, we will discuss the classical Aleksandrov-Fenchel inequalities for quermassintegrals on convex domains and pose the problem of how to extend the inequalities to non-convex domains. We will survey some recent progress on the problem, then report some of our joint work 9] in which we generalize the k-th stage of the inequalities to a class of (k + 1)-convex domains. Our proof, following the earlier work of Castillion 8] for k =  1 case of the inequalities, uses the method of optimal transport.