The classical Minkowski problem consists in finding a convex polyhedron from their normals to the faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary as a function of the unit normal. The proof of this problem consists of three parts: existence, uniqueness and regularity. 

 

In this talk, we study a Minkowski problem for certain measure, called p-capacitary surface area measure, associated to a compact convex set $E$ with nonempty interior and its p-harmonic capacitary function. If $\mu_E$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel positive measure $\mu$ on the unit sphere, find necessary and sufficient conditions for which there exists a convex body E with $\mu_E = \mu$. We will discuss existence, uniqueness, and regularity of this problem which have a deep connection with the Brunn-Minkowski inequality for p-capacity and Monge-Ampere equation.