p-convexity, p-plurisubharmonicity and the Levi problem

TL;DR

This paper explores advanced concepts in p-convex geometry, establishing key results including a Levi problem analogue, properties of minimal p-dimensional currents, and characterization of extreme rays in p-positive matrices, with broad applications.

Contribution

It introduces new theorems linking local and global p-convexity, analyzes minimal p-currents, and characterizes extreme rays in p-positive matrices, advancing the understanding of p-convex structures.

Findings

Local p-convexity implies global p-convexity.

Support of minimal p-currents is contained in the p-hull of boundary union and core.

Extreme rays in p-positive matrices are characterized.

Abstract

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 p-hull of the boundary union with the "core" of the space. Lastly, the exteme rays in the convex cone of p-positive matrices are characterized. This is a basic result with many applications.