Entropy of convex hulls—some Lorentz norm results

Abstract

AbstractLet A be a subset of a type p Banach space E, 1<p⩽2, such that its entropy numbers satisfy (εn(A))n∈ℓq,s for some q,s∈(0,∞). We show (en(acoA))n∈ℓr,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∈ℓq,s implies (en(acoA))n∈ℓp′,s for all 0<s<∞ and q defined by 1/q=1/p′+1/s