Wheeling around von Neumann–Jordan constant in Banach spaces

Abstract

Abstract. In recent times, many constants in Banach spaces have been defined and/or studied. Relations and inequalities among them (sometimes very complicated) have been indicated. But not much effort has been devoted to organize all connections, also because the literature on the subject is growing at an always bigger rate. Here we give some new connections which better the insight on some of them. In particular, we improve a known inequality between the von Neumann-Jordan and James constants.

Some inequalities concerning the James constant in Banach spaces

“…The von Neumann-Jordan constant C NJ (X) was first introduced by Clarkson [5] and reformulated as the above form by Kato, Maligranda and Takahashi in [10]. The last one was first introduced by Gao in [7] and used with the above notation by Alonso, Martín and Papini in [2]. Now let us collect some useful properties concerning these constants (see for example [2][3][4]8]):…”

“…The last one was first introduced by Gao in [7] and used with the above notation by Alonso, Martín and Papini in [2]. Now let us collect some useful properties concerning these constants (see for example [2][3][4]8]):…”

“…We would also like to thank Professor Papini for having provided us the electric version of the paper [2].…”

A note on Jordan–von Neumann constant and James constant

“…Many recent studies have focused on the von NeumannJordan (NJ) constant and James constant (cf. [1][2][3][4][5][6][7][8][9][10][11][12][13]). The constant J (X) = sup min x + y , x − y : x, y ∈ S X is called the non-square or James constant of X .…”

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

“…There are also interesting connections between C NJ (X) and other geometrical constants (see e.g. [2]). …”