The skew generalized Von Neumann Jordan constant in the unit sphere

Abstract

In this paper, we introduce a new constant for Banach spaces, denoted as ${\tilde{C}}_{\mathrm{NJ}}^p(\xi ,v,X)$. We provide calculations for both the lower and upper bounds of this constant, as well as its exact values in certain Banach spaces. Furthermore, we give the inequality relationship between the ${\tilde{C}}_{\mathrm{NJ}}^p(\xi ,v,X)$ constant and the other two constants. Besides, we establish an equivalent relationship between the ${\tilde{C}}_{\mathrm{NJ}}^p(\xi ,v,X)$ constant and the ${\tilde{C}}_{\mathrm{NJ}}^{(p)}(X)$ constant. Specifically, we shall exhibit the connections between the constant ${\tilde{C}}_{\mathrm{NJ}}^p(\xi ,v,X)$ and certain geometric characteristics of Ba nach spaces, including uniform convexity and uniform nonsquareness. Additionally, a sufficient condition for uniform normal structure about the ${\tilde{C}}_{\mathrm{NJ}}^p(\xi ,v,X)$ constant is also established.

arXiv comment19 pages

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