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

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