On a convex level set of a plurisubharmonic function and the support of the Monge-Amp\`ere current

TL;DR

This paper investigates the geometric properties of continuous plurisubharmonic functions solving the Monge-Ampère equation with convex level sets, establishing a relation between Monge-Ampère supports and complex extremal points in convex domains.

Contribution

It introduces a new minimum principle for maximal plurisubharmonic functions and links Monge-Ampère current supports to complex extremal points in convex analysis.

Findings

Established a minimum principle for maximal plurisubharmonic functions.

Connected Monge-Ampère current supports with complex k-extreme points.

Extended results using Lempert's theorems in convex domains.

Abstract

In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Amp\`ere equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal plurisubharmonic function. By using our results and Lempert's results, we show a relation between the supports of the Monge-Amp\`ere currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $\mathbb{C}^{n}$.