Stability of diametral diameter two properties

We prove that the diametral diameter two properties are inherited by $F$-ideals (e.g., $M$-ideals). On the other hand, these properties are lifted from an $M$-ideal to the superspace under strong geometric assumptions. We also show that all of the diametral diameter two properties are stable under the formation of corresponding K"othe-Bochner spaces (e.g., $L_p$-Bochner spaces). Finally, we investigate when the projective tensor product of two Banach spaces has some diametral diameter two property.