P-convexity, p-plurisubharmonicity and the Levi Problem

Three results in p-convex geometry are established. First is the analogue of the Levi problem in several complex variables: namely, local p-convexity implies global p-convexity. The second asserts that the support of a minimal p-dimensional current is contained in the union of the p-hull of the boundary with the "core" of the space. Lastly, the extreme rays in the convex cone of p-positive matrices are characterized. This is a basic result with many applications.