This is an announcement for the paper "Extreme differences between weakly open subsets and convex of slices in Banach spaces" by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda Zoca. Abstract: 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. Archive classification: math.FA Submitted from: glopezp@ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1309.4950 or http://arXiv.org/abs/1309.4950