Slicely countably determined Banach spaces

Abstract

A (separable) Banach space X is slicely countably determined if for every convex bounded subset A of X there is a sequence of slices ( S n ) such that each slice of A contains one of the S n . SCD-spaces form a joint generalization of spaces not containing ℓ 1 and those having the Radon–Nikodým property. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. To cite this article: A. Avilés et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).