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 is an element of (1; 2], we prove that for every beta is an element of [0; 1) and all n is an element of N sup k(1/p') (log k)(beta -1) e(k)(aco A) less than or equal to c sup k(1/p') (log k)(beta) e(k)(A) k less than or equal ton k less than or equal ton holds, where aco A 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 beta is an element of (-infinity, 1) with e(n)(A) less than or equal to n(-1/p')(log n)(-beta) for all n is an element of N the inequality e(n)(aco A) less than or equal to c n(-1/p')(log n)(-beta +1) holds true for all n is an element of N. We also prove that this estimate is asymptotically optimal whenever E has no better type than p. For beta = 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.