Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega$ of dimension $n$. We denote by $H^{\infty }\left(D\right)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_K^D$ a compact subset of the space $C\left(K\right)$ consisted of all restrictions of functions from the unit ball ${𝔹}_{H^{\infty }\left(D\right)}$. In 1950ies Kolmogorov posed a problem: does$${\mathscr{H}}_{\epsilon }\left(A_K^D\right)\sim \tau {\left(ln\frac{1}{\epsilon }\right)}^{n+1},\epsilon \to 0,$$where ${\mathscr{H}}_{\epsilon }\left(A_K^D\right)$ is the $\epsilon$-entropy of the compact $A_K^D$. We give here a survey of results concerned with this problem and a related problem on the strict asymptotics of Kolmogorov diameters...

A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those determined by generic holomorphic mappings) in terms of decomposability of certain convex sets in ℝⁿ. Another class of extreme singularities is presented by means of a notion of relative type.