Some Properties Concerning the JL(X) and ΥJ(X) Which Related to Some Special Inscribed Triangles of Unit Ball

In this paper, we will make some further discussions on the J(L)(X) and Upsilon(J) ( X) which are symmetric and related to the side lengths of some special inscribed triangles of the unit ball, and also introduce two new geometric constants L-1 (X, Delta), L-2 (X, Delta) which related to the perimeters of some special inscribed triangles of the unit ball. Firstly, we discuss the relations among J(L) (X), Upsilon(J) (X) and some geometric properties of Banach spaces, including uniformly non-square and uniformly convex. It is worth noting that we point out that uniform non-square spaces can be characterized by the side lengths of some special inscribed triangles of unit ball. Secondly, we establish some inequalities for J(L) (X), Upsilon(J) ( X) and some significant geometric constants, including the James constant J (X) and the von Neumann-Jordan constant C-NJ(X). Finally, we introduce the two new geometric constants L-1 (X, Delta), L-2 (X, Delta), and calculate the bounds of L-1 (X, Delta) and L-2(X, Delta) as well as the values of L-1 (X, Delta) and L-2 (X, Delta) for two Banach spaces.