How orthogonality influences geometric constants

TL;DR

This paper explores how orthogonality affects geometric constants in Banach spaces, revealing equivalences among several constants when restricted to the unit sphere and proposing a broader insight into their interrelations.

Contribution

It establishes equivalences among four geometric constants based on isosceles orthogonality, linking their definitions on the entire space to their restrictions on the unit sphere.

Findings

Four constants are equivalent under isosceles orthogonality.

Constants restricted to the unit sphere are interconnected.

Insight into how orthogonality influences geometric constants.

Abstract

In this paper, based on isosceles orthogonality, we have found equivalent definitions for four constants: $A_2(X)$ proposed by Baronti in 2000 [J. Math. Anal. Appl. 252(2000), 124-146], $C'_{\mathrm{NJ}}(X)$ introduced by Alonso et al. in 2008 [Stud. Math. 188(2008), 135-150], $T(X)$ introduced by Alonso et al. in 2008 [J. Math. Anal. Appl. 340(2008), 1271-1283] and $L'_{\mathrm{YJ}}(X)$ put forward by Liu et al. in 2022 [Bull. Malays. Math. Sci. Soc., 45(2022),307-321]. A core commonality among these three constants is that they are all restricted to the unit sphere. This finding provides us with an insight: could it be that several constants defined over the entire space, when combined with orthogonality conditions, are equivalent to being restricted to the unit sphere?