Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants

In this paper, we firstly investigate the constant H(X) proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant GL(X) closely related to H(X), along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms GL(X,p) and CL(X) explicitly.

On Geometric Constants for Discrete Morrey Spaces

In this paper we prove that the n-th Von Neumann-Jordan constant and the n-th James constant for discrete Morrey spaces lpq where 1≤p<q<∞ are both equal to n. This result tells us that the discrete Morrey spaces are not uniformly non-l1, and hence they are not uniformly n-convex.

Some Aspects of Geometric Constants in Modular Spaces

In this paper, we generalize the typical geometric constants of Banach spaces to modular spaces. We study the equivalence between the convexity of modular and normed spaces, and obtain the relationship between ρ-Neumann-Jordan constant and ρ-James constant. In particular, we extend the convexity and smoothness modular, and obtain the criterion theorems of the uniform convexity and strict convexity.

Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

We introduce a new geometric constant Jin(X) based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, Jin(X) becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.

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

In this paper, we will make some further discussions on the JL(X) and YJ(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 L1(X,▵), L2(X,▵) which related to the perimeters of some special inscribed triangles of the unit ball. Firstly, we discuss the relations among JL(X), YJ(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 JL(X), YJ(X) and some significant geometric constants, including the James constant J(X) and the von Neumann-Jordan constant CNJ(X). Finally, we introduce the two new geometric constants L1(X,▵), L2(X,▵), and calculate the bounds of L1(X,▵) and L2(X,▵) as well as the values of L1(X,▵) and L2(X,▵) for two Banach spaces.

New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants JL(X) and YJ(X) in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of JL(X) and YJ(X) as well as the values of JL(X) and YJ(X) for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for JL(X), YJ(X) and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established.

On a New Geometric Constant Related to the Euler-Lagrange Type Identity in Banach Spaces

In this paper, we will introduce a new geometric constant LYJ(λ,μ,X) based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and LYJ(λ,μ,X). Also, this new coefficient is computed for X being concrete space.