p-convexity, p-plurisubharmonicity, and the Levi problem F. HarveyH. Lawson, Jr. 32F1032U0535J7058J32p-convexityp-plurisubhamonicityDirichlet problemLevi 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.4886 10.1512/iumj.2013.62.4886 en Indiana Univ. Math. J. 62 (2013) 149 - 169 state-of-the-art mathematics http://iumj.org/access/