Critically large entropy of absolutely convex hulls in Banach spaces of type pp

Given a precompact subset AA of a Banach space of type pp, we investigate and quantify in which way the rate of decay of the entropy numbers of AA affects the rate of decay of the entropy numbers of the real absolutely convex hull of AA. This problem has been studied by various researchers in different settings during the last 25 years. We get new and best possible results both in the case of logarithmic decay of entropy numbers of AA and in the case that the entropy numbers of AA belong to some Lorentz sequence space making a major step to completing the picture.