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 four constants is that they are all restricted to the unit sphere. This finding provides us with the following insight: could it be that several constants defined over the whole space, when combined with suitable orthogonality conditions, are equivalent to their restrictions to the unit sphere? Motivated by this question, we further study the corresponding problem for Birkhoff-James orthogonality. Because this orthogonality is generally non-symmetric, a direct replacement of isosceles orthogonality is not possible. We therefore introduce a norming-functional rectification method, which represents unit-sphere configurations by rectified Birkhoff-James orthogonal data. Consequently, exact Birkhoff-James orthogonal representations are obtained for the above four constants.