  We show that every Banach space containing isomorphic copies of $c_0$ can be
equivalently renormed so that every nonempty relatively weakly open subset of
its unit ball has diameter 2 and, however, its unit ball still contains convex
combinations of slices with diameter arbitrarily small, which improves in a
optimal way the known results about the size of this kind of subsets in Banach
spaces.
