Weak operator Daugavet property and weakly open sets in tensor product spaces

We obtain new progresses about the diameter two property and the Daugavet property in tensor product spaces. Namely, the main results of the paper are: -If $X^{\ast }$ has the WODP, then $X{\widehat{\otimes }}_{\epsilon }Y$ has the DD2P for any Banach space $Y$. -If $X$ has the WODP, then $X{\widehat{\otimes }}_{\pi }Y$ has the DD2P for any Banach space $Y$. -If $X^{\ast }$ and $Y^{\ast }$ have the WODP then $X{\widehat{\otimes }}_{\epsilon }Y$ has the Daugavet property. The above improve many results in the literature and establish progresses on some open questions.