Geometric mean and triangles inscribed in a semicircle in Banach spaces

Abstract

We consider the triangles with vertices x, −x and y where x, y are points on the unit sphere of a normed space. Using the geometric means of the variable lengths of the sides of these triangles, we define two geometric constants for Banach spaces. These constants are closely related to the modulus of convexity of the space under consideration, and they seem to represent a useful tool to estimate the exact values of the James and Jordan-von Neumann constants of some Banach spaces.

Hölder’s means and fixed points for multivalued nonexpansive mappings

“…for a real number p, by considering Hölder's means. It is obvious that A 2 (X) = H 1 (X) (see [3]) and T(X) = H 0 (X) (see [1]), that is,…”

“…The constant A 2 (X) was defined by Baronti, Casini and Papini in [3] by considering the arithmetic mean of x + y and x − y and the constant T(X) was introduced by Alonso and Llorens-Fuster in [1] by considering the geometric mean between x + y and x − y . It is noteworthy that X is uniformly non-square if and only if H p (X) < 2 (see [8]).…”

“…It is noteworthy that X is uniformly non-square if and only if H p (X) < 2 (see [8]). Now, let us collect some useful properties concerning these constants (see [1,3,28]):…”

Some inequalities concerning the James constant in Banach spaces

“…Let X be the 2 − 1 space. It has been shown that for this space A 2 (X) = 1 + 1/ √ 2 (see [1,Example 25]) and J (X) = √ 8/3 (see [10,Example 2]). Thus…”

An inequality between the James type constant and the modulus of smoothness

“…Therefore, J(X ) = J X ,−∞ (1). It is obvious that the James type constants include some known constants, such as Alonso-Llorens-Fuster's constant T (X) [4], Baronti-Casini-Papini's constant A 2 (X) [5], Gao's constant E(X ) [6] and Yang-Wang's modulus γ X (t) [7]. These constants are defined by T (X) = J X ,0 (1), A 2 (X) = J X ,1 (1), E(X ) = 2J 2 X ,2 (1) and γ X (t) = J 2 X ,2 (t).…”