The super Alternative Daugavet property, unconditional bases and SCD geometry

We answer negatively Question 6.4 posed by Langemets, Lõo, Martín, Perreau and Rueda Zoca concerning the existence of infinite-dimensional Banach spaces with a 1-unconditional basis satisfying the super Alternative Daugavet property.

We also address two recent questions posed by Lõo and Perreau concerning weak topological structures and slicely countably determined phenomena in Banach spaces with unconditional bases.

More precisely, we prove that every bounded convex subset of a Banach space with a Schauder basis that is shrinking or boundedly complete admits a countable weak $\pi$-base, yielding a partial positive answer to Question 5.1.

Finally, we prove that, for every $k>1$, there exists a Banach space with a $k$-unconditional basis whose unit ball is not slicely countably determined, thereby giving a positive answer to Question 5.4.