  We prove that, given two Banach spaces $X$ and $Y$ and bounded, closed convex
sets $C\subseteq X$ and $D\subseteq Y$, if a nonzero element $z\in
\overline{\mathrm{co}}(C\otimes D)\subseteq X\widehat{\otimes}_\pi Y$ is a
preserved extreme point then $z=x_0\otimes y_0$ for some preserved extreme
points $x_0\in C$ and $y_0\in D$, whenever $K(X,Y^*)$ separates points of $X
\widehat{\otimes}_\pi Y$ (in particular, whenever $X$ or $Y$ has the compact
approximation property). Moreover, we prove that if $x_0\in C$ and $y_0\in D$
are weak-strongly exposed points then $x_0\otimes y_0$ is weak-strongly exposed
in $\overline{\mathrm{co}}(C\otimes D)$ whenever $x_0\otimes y_0$ has a
neighbourhood system for the weak topology defined by compact operators.
Furthermore, we find a Banach space $X$ isomorphic to $\ell_2$ with a
weak-strongly exposed point $x_0\in B_X$ such that $x_0\otimes x_0$ is not a
weak-strongly exposed point of the unit ball of $X\widehat{\otimes}_\pi X$.
