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

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 $\mathbb{C}^{n}$.