Reflexive Cones

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of \ell_{1}. Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in a Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the Schur spaces by the properties of reflexive cones is given.Comment: 23 page

CONES LOCALLY ISOMORPHIC TO THE POSITIVE CONE OF L-1(GAMMA)

AbstractWe give necessary and sufficient conditions in order for an infinite-dimensional, closed cone P of a Banach space X to be locally isomorphic to the positive cone l+1(Γ) of l1(Γ)

Embeddability of L1 (μ) in dual spaces, geometry of cones and a characterization of c0

AbstractIn this article we suppose that (Ω,Σ,μ) is a measure space and T an one-to-one, linear, continuous operator of L1(μ) into the dual E′ of a Banach space E. For any measurable set A consider the image T(L+1(μA)) of the positive cone of the space L1(μA) in E′, where μA is the restriction of the measure μ on A. We provide geometrical conditions on the cones T(L+1(μA)) which yield that the measure μ is atomic, i.e., that L1(μ) is lattice isometric to ℓ1(A), where A denotes the set of atoms of μ. This result yields also a new characterization of c0(Γ)

Demand functions and reflexivity

AbstractIn the theory of ordered spaces and in microeconomic theory two important notions, the notion of the base for a cone which is defined by a continuous linear functional and the notion of the budget set are equivalent. In economic theory the maximization of the preference relation of a consumer on any budget set defines the demand correspondence which at any price vector indicates the preferred vectors of goods and this is one of the fundamental notions of this theory. Contrary to the finite-dimensional economies, in the infinite-dimensional ones, the existence of the demand correspondence is not ensured. In this article we show that in reflexive spaces (and in some other classes of Banach spaces), there are only two classes of closed cones, i.e. cones whose any budget set is bounded and cones whose any budget set is unbounded. Based on this dichotomy result, we prove that in the first category of these cones the demand correspondence exists and that it is upper hemicontinuous. We prove also a characterization of reflexive spaces based on the existence of the demand correspondences