Relations between generalized von Neumann-Jordan and James constants for quasi-Banach spaces

Abstract

Abstract Let C N J ( B ) $\mathcal{C}_{NJ} ( \mathcal{B} )$ and J ( B ) $J ( \mathcal{B} )$ be the generalized von Neumann-Jordan and James constants of a quasi-Banach space B $\mathcal{B}$ , respectively. In this paper we shall show the relation between C N J ( B ) $\mathcal {C}_{NJ} ( \mathcal{B} )$ , J ( B ) $J ( \mathcal{B} )$ , and the modulus of convexity. Also, we show that if B $\mathcal{B}$ is not uniform non-square then J ( B ) = C N J ( B ) = 2 $J ( \mathcal{B} )=\mathcal{C}_{NJ} ( \mathcal{B} )=2$ . Moreover, we give an equivalent formula for the generalized von Neumann-Jordan constant.

Keywords

• the generalized von Neumann-Jordan constant
• James constant
• uniform non-squareness
• quasi-normed spaces