  Let $X$ be a real Banach space. A subset $B$ of the dual unit sphere of $X$
is said to be a boundary for $X$, if every element of $X$ attains its norm on
some functional in $B$. The well-known Boundary Problem originally posed by
Godefroy asks whether a bounded subset of $X$ which is compact in the topology
of pointwise convergence on $B$ is already weakly compact. This problem was
recently solved by H.Pfitzner in the positive. In this note we collect some
stronger versions of the solution to the Boundary Problem, most of which are
restricted to special types of Banach spaces. We shall use the results and
techniques of Pfitzner, Cascales et al., Moors and others.
