New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces (Q2135435)

For a domain \(D\subset\mathbb C^n\) and a compact \(K\subset D\), let \(\mathcal A_K^D:=\mathcal O(D,\overline{\mathbb D})|_K\); \(\mathcal A_K^D\) is a compact subset of the Banach space \(\mathcal C(K)\). Define the \textit{Kolmogorov \(m\)-width} \[ d_m:=\inf\{\sup\{\inf\{\|f-g\|: g\in L\}: f\in\mathcal A_K^D\}: L \text{ is a subspace of } \mathcal C(K),\; \dim L<m\}. \] In the case where \(D\) is strictly hyperconvex and \(K\) is non-pluripolar, the authors present a new proof of the asymptotic equality \[ \lim_{m\to+\infty}\frac{-\log d_m}{m^{1/n}}=2\pi(\frac{n!}{C(K,D})^{1/n}, \] where \(C(K,D):=\sup\{\int_K(dd^cu)^n: u\in\mathcal{PSH}(D,(-1,0))\}\) is the relative capacity. The proof does not rely on Zakharyuta's Conjecture.