The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Ampère equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and sufficient condition for existence and uniqueness) in dimension 2 is presented. In higher dimensions, partial results are demonstrated.