Tingley’s problem on symmetric absolute normalized norms on ℝ 2

In this paper,we study Tingley＇s problem on symmetric absolute normalized norms on R^2.We construct new methods for Tingley＇s problem on two-dimensional spaces by using isosceles orthogonality,which does not make use of the notion of natural extension.Furthermore,using our methods,several sufficient conditions for Tingley＇s problem on symmetric absolute normalized norms on R2 are given.As applications,we present various new examples including the two-dimensional Lorentz sequence space d^（2）（ω,q） and its dual d^（2）（ω,q）＊by simple arguments.