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

Julio Becerra Guerrero, Ginés López-Pérez, Abraham Rueda Zoca

Published 2013-09-19, updated 2014-10-16Version 3

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.