Some remarks on the geometry of Lorentz spaces
The author considers several properties of Lorentz spaces \(\Lambda _{1, w}\). Section 2 deals with the Kadec-Klee property with respect to local convergence in measure and several related notions. It is worth mentioning that Kadec-Klee properties with respect to local (global) convergence in measure in Calderón-Lozanovskiĭ spaces have recently been studied in [\textit{P. Kolwicz}, ``Kadec-Klee properties of Calderón-Lozanovskiĭ function spaces'', J. Funct. Spaces Appl. 2012, Article ID 314068, 21 p. (2012; Zbl 1244.46012)] and [\textit{P. Kolwicz}, ``Kadec-Klee properties of Calderón-Lozanovskiĭ sequence spaces'', Collect. Math. 63 , 45--58 (2012)]. In Section 3 the author characterizes non-square Lorentz spaces \(\Lambda _{1, w}\) and he also shows that these spaces are not locally uniformly non-square. In the last section extreme points of Lorentz spaces \(\Lambda _{1, w}\) are considered and described in full. Moreover, it is proved that these spaces have no strong U-points (recall that each strong U-points is an extreme point). The results in Sections 3 and 4 require quite delicate proofs. The author applies methods concerning decreasing rearrangements and symmetric spaces which may be of independent interest.
scientific article; zbMATH DE number 6070488
scientific article