  We prove that given a locally compact Hausdorff space, $K$, and a compact
C$^*$-algebra, $\mathcal{A}$, the C$^*$-algebra $C(K, \mathcal{A})$ satisfies
that every convex combination of slices of the closed unit ball is relatively
weakly open subset of the closed unit ball if and only if $K$ is scattered and
$\mathcal{A}$ is the $c_0$-sum of finite-dimensional C$^*$-algebras. We
introduce and study Banach spaces which have property $(\overline{\hbox{P1}})$,
i. e.
  For every convex combination of slices $C$ of the unit ball of a Banach space
$X$ and $x\in C$ there exists $W$ relatively weakly open set containing $x$,
such that $W\subseteq \overline{C}$.
  In the setting of general C$^*$-algebras we obtain a characterization of this
property. Indeed, a C$^*$-algebra has property $(\overline{\hbox{P1}})$ if and
only if is scattered with finite dimensional irreducible representations. Some
stability results for Banach spaces satisfying property
$(\overline{\hbox{P1}})$ are also given. As a consequence of these results we
prove that a real $L_1$-predual Banach space contains no isomorphic copy of
$\ell_1$ if and only if it has property $(\overline{\hbox{P1}})$.
