Entropy of convex hulls -- some Lorentz norm results (Q596803)

Let \(A\) be a precompact subset of a Banach space \(X\) of type \(1< p\leq 2\) such that its entropy numbers satisfy \((\varepsilon_n (A))^\infty_{n=1} \in l_{qs}\) for some \(q,s\in(0,\infty)\). It is proved that the dyadic entropy numbers (i.e., \(e_n: =\varepsilon_{2^n})\) of the absolute convex hull \(acoA\) of \(A\) belongs to Lorentz space \(l_{rs}\) with \(r\) defined by \(1/r=1-1/p+1/q\). This and the second main estimates are consequences of a general inequality relating \((e_n(aco A))^\infty_{n=1}\) with \((\varepsilon_n(A))^\infty_{n=1}\).