On nonlinear $n$-widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear $n$-width $\delta _n(W,X)$ of a subset $W$ in a normed linear space $X$. We proved by a topological method that for $\delta _n(W,X)$ and the well-known Aleksandrov $n$-width $a_n(W,X)$ in a Banach space $X$ the following inequalities hold: $\delta _{2n+1}(W,X)\le a_n(W,X)\le \delta _n(W,X)$. Let $K_{p,\theta }^{\alpha }$ be the unit ball of Besov space $B_{p,\theta }^{\alpha },\quad \alpha >0,\quad 1\le p,\theta \le \infty$, of multivariate periodic functions. Then for approximation in $L_q,\quad 1\le q\le \infty$, with some restriction on $p,q$ and $\alpha$, we established the asymptotic degree of these $n$-widths: $a_n(K_{p,\theta }^{\alpha },L_q)\approx \delta _n(K_{p,\theta }^{\alpha }, L_q)\approx n^{-\alpha }$.