On Lelong-Bremermann Lemma

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma:

Let $u$ be a continuous plurisubharmonic function on a Stein manifold $\Omega$ of dimension $n.$ Then there exists an integer $m\leq 2n+1$, natural numbers $p_{s}$, and analytic mappings $G_{s}=\left( g_{j}^{\left( s\right) }\right) :\Omega \rightarrow \mathbb{C}^{m}, s=1,2,...,$ such that the sequence of functions

$$u_{s}\left( z\right) =\frac{1}{p_{s}}\max \left( \ln \left\vert g_{j}^{\left( s\right) }\left( z\right) \right\vert :\text{ }j=1,\ldots ,m\right)$$

converges to $u$ uniformly on each compact subset of $\Omega$.

In the case when $\Omega$ is a domain in the complex plane, it is shown that one can take $m=2$ in the theorem above (Section 3); on the other hand, for $n$-circular plurisubharmonic functions in $\mathbb{C}^{n}$ the statement of this theorem is true with $m=n+1$ (Section 4). The last section contains some remarks and open questions.