Abstract: Given a domain D in C^n and K a compact subset of D, we denote A_K^D the compact set in C(K), of all restrictions in K of holomorphic functions on D bounded by 1. The sequence (d_m(A_K^D))_{m\in N} of Kolmogorov m-widths of A_K^D provides a measure of the degree of compactness of the set A_K^D in C(K) and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on $\epsilon$-entropy of compact sets in the 1950s. This problem has already been proved in 2004 by S.N., using pluripotential theory technics. Here, with O. Bandtlow, we give a totally new proof of these asymptotics. We proceed by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results for the eigenvalues of a certain family of Toeplitz operators, defined on a family of Bergman spaces. While the upper bounds follow from an application of the Bergman-Weil formula together with an exhaustion procedure by special holomorphic polyhedra.