Big slices versus big relatively weakly open subsets in Banach spaces

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 c0 isomorphically can be equivalently renormed so that every slice of its unit ball has diameter 2 and still its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small, which answers an open question and stresses the differences between the size of slices and relatively weakly open subsets of the unit ball of Banach spaces.