Generalized Modulus Of Convexity And A New Characteristic Function

The geometry theory of Banach space involves many contents,a wide range of fields and all aspects.Therefore,the geometric properties of its space have always been the focus of the research of pure mathematics workers engaged in Banach spaces geometry.In recent years,related mathematicians can always produce new ideas and gratifying achievements in the exploration of geometric structure,one of the hottest topics is based on modulus of convexity and modulus of smoothness,starting from the properties and characteristics of modulus of convexity and modulus of smoothness,and combining with modulus of convexity and modulus of smoothness by introducing other related knowledge in Banach space.Thus,the innovative achievements in the geometric structure of Banach space are obtained.To a great extent,this has aroused the strong research interest of mathematics enthusiasts engaged in geometric theory,and provideds enlightenment and guidance for the deficiency of the related content of Banach space geometry theory.In this paper,from the point of view of generalized modulus of convexity,focusing on the definition of generalized modulus of convexity,we begin to describe the related properties of geometric theory in Banach space.After the conclusion that modulus of convexity characterize the geometry of Banach spaces,we continue to explore whether generalized modulus of convexity have such properties.By combining the generalized modulus of convexity with the knowledge of geometric theory,the relationship between them is reconsidered,and the related properties and conclusions of Banach spaces in uniformly non-square,uniformly convex and uniformly smooth geometry are given by this research idea.The article is divided into three parts.Firstly,in the first part,the basis of topic selection is introduced,and the developmen t process of modulus of convexity and generalized modulus of convexity is introduced.According to references at home and abroad.It also introduces the main contents of this paper and the methods applied in this study,and formulates the main research framework for this paper.In the second part,using the definition and properties of generalized modulus of convexity,starting from the related conclusions of modulus of convexity in uniformly nonsquare space,we continue to explore whether generalized modulus of convexity has these properti es and conclusions.Include whether generalize modulus of convexity have new theoremes for judging uniformly nonsquare space,the relationship between line segments on the unit sphere which is not strictly convex,and considering lp(Xi)(1<p<+∞)and its subspace {Xi},the necessary and sufficient conditions for the generalized modulus of convexity to be uniformly non-square are obtained.In the third part,after studying the geometric properties of a new class of characteristic functions in Banach spaces,continue to explore the combination of generalized modulus of convexity and generalized modulus of smoothness with this new characteristic function,the equivalent relationship between generalized modulus of convexity,generalized modulus of smoothness and this kind of new characteristic function is obtained.A new method for judging uniformly convex and umiformly smooth Banach spaces is given.