Gelfand numbers and metric entropy of convex hulls in Hilbert spaces

We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets K ⊂ H of a Hilbert space H by the metric entropy of the set K where the covering numbers N(K, ") of K by "-balls of H satisfy the Lorentz condition ∫ ∞ 0 ( log2N(K, ") )r/s d" <∞ for some fixed 0 < r, s ≤ ∞ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 < r < ∞, s = ∞. Moreover, we obtain optimal estimates of Gelfand numbers of absolutely convex hulls if the metric entropy satisfies the entropy condition sup ">0 " ( log2N(K, ") )1/r( log2(2 + log2N(K, ")) ) <∞ fore some fixed 0 < r <∞,−∞ < <∞. Using inequalities between Gelfand and entropy numbers we also get optimal estimates of the metric entropy of the absolutely convex hull cov(K). As an interesting feature of the estimates, a sudden jump of the asymptotic behavior of Gelfand numbers as well as of the metric entropy of absolutely convex hulls occurs for fixed s if the parameter r crosses the point r = 2 and, if r = 2 is fixed, if the parameter crosses the point = 1. The results established in Hilbert spaces extend and recover corresponding results of several authors. The proofs are based on two inequalities already discovered in [CKP99]. 2010 Mathematics Subject Classification. Primary 47B06, 52A23, 41A46.