  We study smoothness and strict convexity of (the bidual) of Banach spaces in
the presence of diameter 2 properties. We prove that the strong diameter 2
property prevents the bidual from being strictly convex and being smooth, and
we initiate the investigation whether the same is true for the (local) diameter
2 property. We also give characterizations of the following property for a
Banach space $X$: "For every slice $S$ of $B_X$ and every norm-one element $x$
in $S$, there is a point $y\in S$ in distance as close to 2 as we want." Spaces
with this property are shown to have non-smooth bidual.
