  We study the unknown differences between the size of slices and relatively
weakly open subsets of the unit ball in Banach spaces. We show that every
Banach space containing isomorphic copies of $c_0$ can be equivalently renormed
so that every slice of its unit ball has diameter 2 and satisfying that its
unit ball contains nonempty relatively weakly open subsets with diameter
arbitrarily small, which answers an open problem and stresses the differences
of diameter between slices and relatively weakly open subsets of the unit ball
in Banach spaces.
