Entropy of convex hulls—some Lorentz norm results

https://doi.org/10.1016/j.jat.2004.04.001Get rights and content
Under an Elsevier user license
open archive

Abstract

Let 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.

MSC

41A46
46B07
46B20
52A07

Keywords

Entropy numbers
Convex hulls

Cited by (0)