Dependence of Kolmogorov widths on the ambient space (Q2848851)

In this interesting paper the authors study the dependence of the Kolmogorov \(n\)-widths of a compact set on the ambient space. If \(K\) is a (compact) subset of a normed linear space \(Y\), then the Kolmogorov \(n\)-width \(d_n(K, Y)\) is defined by NEWLINE\[NEWLINEd_n(K, Y) = \inf_{L_n} \sup_{x\in K} \inf_{\ell\in L_n} \|x-\ell\|,NEWLINE\]NEWLINE where \(L_n\) varies over all linear subspaces of \(Y\) of dimension at most \(n\). It is known that this value may well depend on \(Y\). That is, if \(X\) is a normed linear space containing \(Y\) as a subspace, then we may have NEWLINE\[NEWLINEd_n(K, Y) > d_n(K, X).NEWLINE\]NEWLINE The absolute Kolmogorov \(n\)-width of \(K\) is defined by NEWLINE\[NEWLINEd_n^a(K) = \inf_X d_n(K, X),NEWLINE\]NEWLINE where the infimum is taken over all normed linear spaces \(X\) containing \(Y\). This paper studies various properties of this \(d_n^a(K)\), gives examples of compact \(K\) with \(d_n(K, Y)=d_n^a(K)\) for any normed linear space \(Y\) containing \(K\), and examples of compact \(K\), and \(Y\), where NEWLINE\[NEWLINE\limsup_{n\to\infty} {{d_n(K, Y)}\over {d_n^a(K)}} =\infty,NEWLINE\]NEWLINE as well as numerous other results such as the relations with other \(n\)-widths, and Kolmogorov \(n\)-widths of images of compacts under compact operators. Seven ``problems'' are presented.