Some remarks on the geometry of Lorentz spaces Λ1,wΛ1,w

Some geometric properties of classical Lorentz spaces Λ1,wΛ1,w are considered. First criteria for the Kadec–Klee property with respect to the local convergence in measure for Lorentz spaces Λ1,wΛ1,w are given. In order to prove these criteria it was necessary to find first weaker sufficient conditions for the almost everywhere convergence of a sequence of rearrangements (xn∗) to a rearrangement element x∗x∗. Next criteria for non-squareness as well as for extreme points of the unit ball of the spaces are established. The last result is a generalization of the result presented in Carothers et al. (1992) [5].