On attaining diameter two and some related properties in Banach spaces

We study versions of the diameter two properties where the diameter of slices, weakly relatively open subsets, or convex combinations of slices of the unit ball is attained. We also introduce some new related properties in Banach spaces and show connections between them. We show that among all Musielak–Orlicz spaces over a complete σ-finite measure space the only locally square spaces are the L1 spaces over a non-atomic measure space.