Metric entropy of convex hulls in type p spaces-the critical case
Given a precompact subset A of a type p Banach space E, where p ∈ (1, 2], we prove that for every β E [0,1) and all n ∈ N sup k 1/p' (log k) β-1 e k (aco A) ≤ c sup k 1/p' (log k) β e k (A) k≤n holds, where acoA is the absolutely convex hull of A and e k (.) denotes the k th dyadic entropy number. With this inequality we show in particular that for given A and β ∈ (-∞,1) with e n (A) ≤ n -1/p' (log n) -β for all n ∈ N the inequality e n (aco A) < c n -1/p' (log n) -β+1 holds true for all n E N. We also prove that this estimate is asymptotically optimal whenever E has no better type than p. For β = 0 this answers a question raised by Carl, Kyrezi, and Pajor which has been solved up to now only for the Hilbert space case by F. Gao.