Kolmogorov problem on widths asymptotics

Given a compact set $K$ in an open set $D$ on a Stein manifold $\Omega ,\dim
\Omega =n,$ the set $A_{K}^{D}$ of all restrictions of functions, analytic
in $D$ with absolute value bounded by $1,$ is considered as a compact subset
in $C\left( K\right) $. The problem about the strict asymptotics for
Kolmogorov diameters: $-\ln d_{s}\left( A_{K}^{D}\right) \sim \sigma \
s^{1/n}$ as $s\rightarrow \infty ,$ was stated by Kolmogorov (in an
equivalent formulation for $\varepsilon $-entropy of this set) in 1950ths.
In our talk we discuss a solution of the Kolmogorov problem, which is a
synthesis of 1) \textit{our result about the reduction of Kolmogorv problem
to the problem of Pluripotential Theory about approximation of Green
pluripotential of the pluricondenser }$\left( K,D\right) $\textit{\ by
pluripotentials with finite set of logarithmic singularities }and 2) the
\textit{recent result of Nivoche and Poletsky, solving the latter problem}.
Some related unsolved problem will be discussed.