Let $T:X\rightarrow Y$ be a linear continuous map between Banach spaces $X$ and $Y$ and denote by $T':Y'\rightarrow X'$ the norm adjoint of $T$. Let $M\subseteq U'$ be a subset of
the unit sphere $U'$ of $Y'$. Let $C\subseteq M$ be any weak*-closed convex subset. Is the image $T'(C)$ weak*-closed in $X'$? Or more generally, assuming that $C$ is convex, under which additional conditions on $C$ is the image $T'(C)$ weak*-closed in $X'$?