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Abstract

For a precompact subset $K$ of a Hilbert space we prove the following inequalities: $$ n^{1/2} c_n(\mathop{\rm cov}\nolimits(K))\le c_K\Big(1+\sum^n_{k=1} k^{-1/2}e_k(K)\Big),\quad\ n\in \mathbb N, $$ and $$ k^{1/2} c_{k+n}(\mathop{\rm cov}\nolimits(K))\le c\bigg[\log^{1/2}(n+1)\varepsilon_n(K)+\sum_{j=n+1}^\infty \frac{\varepsilon_j(K)}{j\log^{1/2}(j+1)}\bigg], $$ $k,n\in\mathbb N$, where $c_n(\mathop{\rm cov}\nolimits(K))$ is the $n$th Gelfand number of the absolutely convex hull of $K$ and $\varepsilon_k(K)$ and $e_k(K)$ denote the $k$th entropy and $k$th dyadic entropy number of $K$, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers $c_n(\mathop{\rm cov}\nolimits(K))$ provided that the entropy numbers $\varepsilon_n(K)$ are slowly decreasing. For example, we get optimal estimates in the non-critical case where $\varepsilon_n(K)\preceq \log^{-\alpha}(n+1)$, $\alpha\not=1/2$, $0<\alpha<\infty$, as well as in the critical case where $\alpha=1/2$. For $\alpha=1/2$ we show the asymptotically optimal estimate $c_n(\mathop{\rm cov}\nolimits(K))\preceq n^{-1/2}\log(n+1)$, which refines the corresponding result of Gao [Ga] obtained for entropy numbers. Furthermore, we establish inequalities similar to that of Creutzig and Steinwart [CrSt] in the critical as well as non-critical cases. Finally, we give an alternative proof of a result by Li and Linde [LL] for Gelfand and entropy numbers of the absolutely convex hull of $K$ when $K$ has the shape $K=\{t_1,t_2,\ldots\}$, where $\|t_n\|\le \sigma_n$, $ \sigma_n\downarrow 0$. In particular, for $\sigma_n\le \log^{-1/2}(n+1)$, which corresponds to the critical case, we get a better asymptotic behaviour of Gelfand numbers, $c_n(\mathop{\rm cov}\nolimits(K))\preceq n^{-1/2}$.