On James and Jordan–von Neumann constants and the normal structure coefficient of Banach spaces

Abstract

Abstract. Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant C NJ (X), and the normal structure coefficient N (X) of Banach spaces X are investigated. Relations between J(X) and J(X * ) are given as an answer to a problem of Gao and Lau [16]. Connections between C NJ (X) and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the C NJ (X)-constant, which implies that a Banach space with C NJ (X)-constant less than 5/4 has the fixed point property. . In particular, they determined or estimated C NJ (X) for various spaces X, and showed that some properties of X such as uniform non-squareness, superreflexivity or type and cotype can be described in terms of the constant C NJ (X).The aim of this paper is to clarify some relations beween the C NJ (X)-constant and some other geometrical constants, especially the non-square constant J(X) of James and the normal structure coefficient N (X). In addition, we investigate the James constant J(X) more carefully. Everything is supported by several examples of concrete Banach spaces with the calculation of these constants.The paper is organized as follows: In Section 1 we collect necessary properties of the modulus of convexity and modulus of smoothness. In Section 2 the uniformly non-square spaces and the James constant J(X) are consid-2000 Mathematics Subject Classification: 46B20, 46E30, 46A45, 46B25.

On Von Neumann–jordan Constants

“…Many results on the constants C N J (X ) andC N J (X ) for various X have been proved by Kato et al [3][4][5][6][7][8]. In particular, Kato and Takahashi [5] showed thatC N J (X ) < 2 if and only if X is superreflexive.…”

The Ptolemy constant of absolute normalized norms on ℝ2

“…There are several constants defined on Banach spaces such as the Gao [1] and von Neumann-Jordan constants [2]. It has been shown that these constants are very useful in geometric theory of Banach spaces, which enable us to classify several important concepts of Banach spaces such as uniformly non-squareness and uniform normal structure [3][4][5][6][7][8].…”

“…Let us restrict ourselves to the 2D Cesàro sequence space ces (2) p which is just ℝ 2 equipped with the norm defined by norm defined by…”

Normed spaces equivalent to inner product spaces and stability of functional equations

“…for example [9,19,20,25]). Generally, one could say that the smaller the C NJ (X) constant is, the better properties the space X possesses.…”