Average \(n\)-widths of the Wiener space in the \(L_ \infty\)-norm (Q1260655)

Let \(w\) denote the Wiener measure on \(C= C[ 0,1]\). The average \(n\)-width of \((C,w)\) in \(L_ q[ 0,1]\) \((1\leq q\leq\infty)\) is defined by \[ d_ n(C,w,L_ q)= \inf_ X \int_ C \inf_{y\in X} \| x-y\|_{L_ q} w(dx), \] where \(X\) runs through all linear subspaces of \(L_ q\) with \(\dim X\leq n\). For \(1\leq q<\infty\) the author proved in two earlier papers that \[ d_ n(C, w,L_ q) \asymp n^{-1/2}. \] In the present paper it is shown that this asymptotic formula is also valid for the case \(q=\infty\).