On geometric structure of symmetric spaces

In this article we discuss local approach to strict K-monotonicity and local uniform rotundity in symmetric spaces. We prove several general results on local structure of symmetric spaces E showing relation between strict monotonicity and strict K-monotonicity and the Kadec–Klee property for global convergence in measure. We also present the full criteria for points of upper K  -monotonicity in Lorentz spaces Γp,wΓp,w for degenerated weight function w. Next we characterize local uniform rotundity in symmetric spaces E   proving several correspondences between x∈Ex∈E a point of local uniform rotundity and its decreasing rearrangement x⁎x⁎ and absolute value |x||x|. Finally, we apply these results to find complete criteria for local uniform rotundity of Lorentz spaces Γp,wΓp,w.