Geometry of Banach spaces with property beta

We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property beta and has the same character of density. Then we show that, nevertheless, property beta satis fies a hereditary property. We study strong subdifferentiability of norms w ith property beta to characterize separable polyhedral Banach spaces as tho se admitting a strongly subdifferentiable beta norm. In general, a Banach s pace with such a norm is polyhedral. Finally, we provide examples of non-re flexive spaces whose usual norm fails property beta and yet it can be appro ximated by norms with this property, namely (L-1[0, 1], parallel to . paral lel to(1)) and (C(K), parallel to . parallel to(infinity)) where K is a sep arable Hausdorff compact space.