Extreme differences between weakly open subsets and convex combinations of slices in Banach spaces

We show that every Banach space containing isomorphic copies of c0c0 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 an optimal way the known results about the size of this kind of subsets in Banach spaces.