This is an announcement for the paper “Almost square dual Banach spaces” by Trond A. Abrahamsenhttps://arxiv.org/search/math?searchtype=author&query=Abrahamsen%2C+T+A, Petr Hájekhttps://arxiv.org/search/math?searchtype=author&query=H%C3%A1jek%2C+P, Stanimir Troyanskihttps://arxiv.org/search/math?searchtype=author&query=Troyanski%2C+S.

Abstract: We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing $c_0$ has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals.

https://arxiv.org/abs/1912.12519