Skew generalized von Neumann-Jordan constant in Banach Spaces

Abstract

We introduce a new geometric constant $C^{p}_{NJ}(\zeta,\eta,X)$ in Banach spaces, which is called the skew generalized von Neumann-Jordan constant. First, the upper and lower bounds of the new constant are given for any Banach space. Then we calculate the constant values for some particular spaces. On this basis, we discuss the relation between the constant $C^{p}_{NJ}(\zeta,\eta,X)$ and the convexity modules $\delta_X(\varepsilon)$, the James constant $J(X)$. Finally, some sufficient conditions for the uniform normal structure associated with the constant $C^{p}_{NJ}(\zeta,\eta,X)$ are established.