The super Alternative Daugavet property for Banach spaces

Abstract We introduce the super Alternative Daugavet property (super ADP), which lies strictly between the Daugavet property (DP) and the ADP. A Banach space X has the super ADP if for every element x in the unit sphere and every relatively weakly open subset W of the unit ball intersecting the unit sphere, there are an element $y\in W$ and a modulus one scalar θ such that $\|x+\theta y\|$ is almost two. Spaces with the DP satisfy this condition, and it implies the ADP. We first provide examples of super ADP spaces that fail the DP. We show that the norm of a super ADP space is rough, hence the space cannot be Asplund, and we also prove that the space fails the point of continuity property (particularly, the Radon–Nikodým property). In particular, we get examples of spaces with the ADP that fail the super ADP. For a better understanding of the differences between the super ADP, the DP and the ADP, we consider the localizations of these properties and prove that they behave rather differently. As a consequence, we provide characterizations of the super ADP for spaces of vector-valued continuous functions and of vector-valued integrable functions.