  We prove that the following three properties for a Banach space are all
different from each other: every finite convex combination of slices of the
unit ball is (1) relatively weakly open, (2) has nonempty interior in relative
weak topology of the unit ball, and (3) intersects the unit sphere. In
particular, the $1$-sum of two Banach spaces does not have property (1), but it
has property (2) if both the spaces have property (1); the Banach space
$C[0,1]$ does not have property (2), although it has property (3).
