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  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  for k = 1 case of the inequalities, uses the method of optimal transport.
All Science Journal Classification (ASJC) codes
- Aleksandrov-Fenchel inequality
- mixed volume
- optimal transport map