Some Geometric Properties Of Orlicz Space With Mazur-Orlicz F-Norm

Geometry of Banach space are important in modern functional analysis.He has had important applications in many areas,such as application of nonlinear operator equation theory,ergodic theory,and cybernetics,information theory,and financial mathematics.As well known,nonsquareness and convexity are important contents of geometric theory in Banach space.In 1936,the concept of uniformly convex in Banach space was introduced by Clarkson.Many scholars have been studied it,and a variety of convexity were introduced,such as strictly convex,uniformly convex,local uniformly convex and so on.Inspired B-convex spaces,in 1964,the definition of uniformly nonsquare in normed linear space was introduced by James.In 1976,another definition of uniformly nonsquare in normed linear space was introduced by Sch?ffer.Particularly,whether or not uniformly nonsquare Banach space has fixed point of nonexpansive mapping has been discussed as an open problem.Until 2005,Jesús García-Falset,Enrique Llorens-Fuster and Eva M.Mazcu?an-Navarro solved the open problem and obtained that uniformly nonsquare Banach space has fixed point property.In 2018,Yuan Cui,Henryk Hudzik,Rados?aw Kaczmarek and Pawe? Kolwicz discussed the monotonicity of Orlicz space that generated by the monotone continuous function equipped with Mazur-Orlicz F-norm.In 2020,Xinran Bai,Yunan Cui and Joanna Ko?czak given criteria that Orlicz Spaces that generated by the monotone function equipped with Mazur-Orlicz F-norm have strictly monotonicity,upper locally uniform monotonicity.It's completely different from the Banach space.This opens a new path to the research work we do.Firstly,we introduce the Orlicz space development situation at home and abroad,and introduce the purpose of this studied.Secondly,the definition of uniformly nonsquareness in Banach spaces is extended to F-normed spaces.Most of the results from this paper concern(uniformly)nonsquareness in the sense of James or in the sense of Sch?ffer in Orlicz spaces equipped with the Mazur-Orlicz F-norm.It is well known that uniform nonsquareness in the sense of Sch?ffer and in the sense of James are equivalent in Banach spaces.We found that uniform nonsquareness in the sense of James and in the sense of Sch?ffer are not equivalent for F-normed spaces in this paper.Criteria for Orlicz spaces equipped with the Mazur-Orlicz F-norm to be nonsquare and uniformly nonsquare in the sense of James or in the sense of Sch?ffer are given.Finally,the criterion for the Orlicz space equipped with Mazur-Orlicz F-norm of strictly convex,uniformly convex,locally uniformly convex is given.