Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces

Abstract

We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index 1 are equivalent. In the class of rearrangement invariant (r.i.) sequence spaces the only examples of spaces with these properties are C 0, l 1 and l. The only lush r.i. separable function space on [0,1] is L 1[0,1]; the same space is the only r.i. separable function space on [0,1] with the Daugavet property over the reals.