Geometry of Banach spaces with property β

Abstract

We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L 1[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact space