On exact values of the Kolmogorov width of compact sets in Hilbert space (Q1810224)

It is well known that the Kolmogorov \(n\)-width of a compact set \(K\) in Hilbert space \(H\) (real or complex) is defined as \[ d_n(K,H)=\inf d(K,a+ L_n), \tag{1} \] where \(a\) is a vector in \(H\), \(L_n\) is an \(n\)-dimensional subspace of \(H\), and \(d(K,a+L_n)\) is the distance between \(K\) and the plane \(a+L_n\): \[ d(K,a+L_n)=\sup_{x\in K}\inf_{y\in L_n}\|x-a-y \|.\tag{2} \] A two-sided bound for the Kolmogorov width of compact sets in Hilbert space, that is, for (1)--(2) is established. Moreover, the Kolmogorov width of a set of equidistant points in real Hilbert space and the 1-width of the continuous Wiener spiral are computed.