Properties And Applications Of Generalized Jordan-von Neumann Type Constant With Parameter A

Since the 1940s,the geometry theory of Banach space has been developing vigorously,and it has important applications in variational problems,fluid mechanics,fixed points and other fields.Geometric constant is an important way to explore the geometric properties of Banach space.The description of abstract Banach space can be given by calculating the exact value of the geometric constant and establishing the relationship between the geometric constants.Therefore,the introduction and application of geometric constant have always been the focus of attention.Based on this,this paper mainly consists of the following parts:Firstly,we introduce a new geometric constant,and indicate the relationship between the constant C(p)-∞(a,X)and the constantC-∞(X),C-∞(a,X)and C(p)-∞(X),respectively.It is pointed out that this constant is a non-decreasing continuous function with the parameter a.It is shown that Banach spaces satisfying C(p)-∞(a,X)<2 which is uniformly non-square.We show the relationship between the generalized James constant and this constant,and prove the relationship between this constant and the geometric constant in the ultrapower space.Then,the(a+2)p/2p-2(2p+ap)and 2 are the lower bound and upper bound,respectively.The exact value of this constant are calculated in space l1-l∞ and l2-l∞ with a=1,respectively.And the exact value of this constant is obtained in space l1-l2 with a=2.In addition,based on the lower bound and properties of C(p)-∞(a,X),we guess that its value is(a+2)p/2p-2(2p+ap)in Hilbert space.Finally,we use constant C(p)-∞(a,X)and J(a,X)to give a criterion for normal structure.It follows that the Banach space with C(p)-∞(1,X)<2 has normal structure.Using this result,two sufficient conditions for constant C(p)-∞(a,X)on normal structure are obtained.The results of the constant C-∞(X)and the constantC-∞(a,X)about normal structure are generalized.