Cones with bounded and unbounded bases and reflexivity

In this paper we prove two characterizations of reflexivity for a Banach space XX. The first one is based on the existence in XX of a closed convex cone with a nonempty interior such that all the bases generated by a strictly positive functional are bounded, while the second one is stated in terms of the nonexistence of a cone such that it has bounded and unbounded bases (both generated by strictly positive functionals) simultaneously. We call such a cone mixed based cone. We study the features of this class of cones. In particular, we show that every cone conically isomorphic to the nonnegative orthant ℓ+1 of ℓ1ℓ1 is a mixed based cone and that every mixed based cone contains a conically isomorphic copy of ℓ+1. Moreover we give a detailed description of the structure of a mixed based cone. This approach allows us to prove some results concerning the embeddings of ℓ1ℓ1 and c0c0 in a Banach space.