Geometry of Banach spaces of functions associated with concave functions

Abstract

Let ( X , Σ , μ ) (X,\Sigma ,\mu ) be a positive measure space, and ϕ \phi be a concave nondecreasing function on R + → R + {R^ + } \to {R^ + } with ϕ ( 0 ) = 0 \phi (0) = 0 . Let N ϕ ( R ) {N_\phi }(R) be the Lorentz space associated with the function ϕ \phi . In this paper a complete characterization of the extreme points of the unit ball of N ϕ ( R ) {N_\phi }(R) is provided. It is also shown that the space N ϕ ( R ) {N_\phi }(R) is not reflexive in all nontrivial cases, thus generalizing a result of Lorentz. Several analytical properties of spaces N ϕ ( R ) {N_\phi }(R) , and their abstract analogues N ϕ ( E ) {N_\phi }(E) , are obtained when E is a Banach space.