Entropy of convex hulls--some Lorentz norm results

Let A be a subset of a type p Banach space E, 1 < p ≤ 2, such that its entropy numbers satisfy ( εn(A) ) n ∈ `q,s for some q, s ∈ (0,∞). We show ( en(acoA) ) n ∈ `r,s for the dyadic entropy numbers of the absolutely convex hull acoA of A, where r is defined by 1/r = 1/p′+1/q. Furthermore, we show for slowly decreasing entropy numbers that ( en(A) ) n ∈ `q,s implies ( en(acoA) ) n ∈ `p′,s for all 0 < s <∞ and q defined by 1/q = 1/p′ + 1/s. AMS classification: 41A46, 46B07, 46B20, 52A07