Gaoyong Zhang - The p-Minkowski problem of polytopes

The p-surface area measure of a convex body is defined as the product of the usual surface area measure and the (1-p)-th power of the support function of the convex body. The p-Minkowski problem states: Given a non-negative measure on the unit sphere, what are the necessary and sufficient conditions so that the given measure is the p-surface area measure of a convex body? We give a solution to the problem for polytopes.