  Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set
$\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the
modulus of which is bounded by $1$ is a compact subset of the Banach space
$C(K)$ of continuous functions on $K$. The sequence
$(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of
$\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set
$\mathcal{A}_K^D$ 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. In the 1980s Zakharyuta showed that for suitable $D$
and $K$ the asymptotics \begin{equation*}
  \lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} =
  2\pi \left ( \frac{n!}{C(K,D)}\right ) ^{1/n}\,, \end{equation*} where
$C(K,D)$ is the Bedford-Taylor relative capacity of $K$ in $D$ is implied by a
conjecture, now known as Zakharyuta's Conjecture, concerning the
approximability of the regularised relative extremal function of $K$ and $D$ by
certain pluricomplex Green functions. Zakharyuta's Conjecture was proved by
Nivoche in 2004 thus settling the asymptotics above at the same time.
  We shall give a new proof of the asymptotics above for $D$ strictly
hyperconvex and $K$ non-pluripolar which does not rely on Zakharyuta's
Conjecture. Instead we proceed more directly by a two-pronged approach
establishing sharp upper and lower bounds for the Kolmogorov widths. The lower
bounds follow from concentration results of independent interest for the
eigenvalues of a certain family of Toeplitz operators, while the upper bounds
follow from an application of the Bergman-Weil formula together with an
exhaustion procedure by special holomorphic polyhedra.
