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We show that for every Banach lattice E failing RNP and not containing co (resp. containing co ) and for every e > 0 there exists a solid convex closed subset D of the unit ball of E, such that dist[ext(Y(E S,E E] > I1e(resp. > -) and such that every slice of D has diameter bigger than 2 e . We also prove that these results are optimal. We apply them to construct rough lattice norms with almost optimal constant on non-Asplund Banach lattices.

Nondentable solid subsets in Banach lattices failing RNP. Applications to renormings