Local structure of symmetrizations E(⁎) with applications

We study the local structure of the separated point x   in the space E(⁎)E(⁎) with respect to properties of its nonincreasing rearrangement x⁎x⁎ in the space E. In particular, we get the characterizations of local structure in the Lorentz and Marcinkiewicz spaces. It is worth to mention that the local structure of a separated point can be applicable to the local best dominated approximation problems in Banach lattices.