Nondentable solid subsets in banach lattices failing rnp.applications to renormings

We show that for every Banach lattice E failing RNP and not containing cq (resp. containing cq) and for every ε > 0 there exists a solid convex closed subset D of the unit ball of E, such thatand such that every slice of D has diameter bigger than 2 −ε.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.