The skew generalized Von Neumann Jordan constant in the unit sphere

TL;DR

This paper introduces a new geometric constant for Banach spaces, explores its bounds, relationships with existing constants, and links to geometric properties like convexity and nonsquareness.

Contribution

It defines the skew generalized Von Neumann Jordan constant, calculates bounds, establishes inequalities and relationships with other constants, and connects it to Banach space geometry.

Findings

Exact values of the constant in certain Banach spaces

Inequality relationships with other constants

Connections to uniform convexity and nonsquareness

Abstract

In this paper, we introduce a new constant for Banach spaces, denoted as $\widetilde{C}_{\mathrm{NJ}}^p(\xi, v, X)$. 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 $\widetilde{C}_{\mathrm{NJ}}^p(\xi, v, X)$ constant and the other two constants. Besides, we establish an equivalent relationship between the $\widetilde{C}_{\mathrm{NJ}}^p(\xi, v, X)$ constant and the $\widetilde{C}_{\mathrm{NJ}}^{(p)}(X)$ constant. Specifically, we shall exhibit the connections between the constant $\widetilde{C}_{\mathrm{NJ}}^p(\xi, v, X)$ and certain geometric characteristics of Ba nach spaces, including uniform convexity and uniform nonsquareness. Additionally, a sufficient condition for uniform normal structure about the $\widetilde{C}_{\mathrm{NJ}}^p(\xi, v, X)$ constant is also established.