Some results on the Dunkl-Williams constant

TL;DR

This paper compiles formulas for the Dunkl-Williams constant in real normed spaces, calculates its values for specific 2D spaces, and explores related orthogonality constants.

Contribution

It provides new explicit calculations of the Dunkl-Williams constant for particular two-dimensional spaces and discusses related orthogonality constants.

Findings

Dunkl-Williams constant for ℓ₂–ℓ₁ is 2√2

Dunkl-Williams constant for a dodecahedral space is 8(2−√3)

Formulas for calculating DW(X) are compiled

Abstract

This paper presents a compilation of various formulas for calculating the Dunkl-Williams constant $DW(X)$ of a real normed linear space. The constant $DW_B(X)$ related to Birkhoff orthogonality is also considered. The value of $DW(X)$ is calculated for several two-dimensional spaces. In particular, it is shown that the Dunk-Williams constant for $\ell_2-\ell_1$ is equal to $2\sqrt{2}$, and that it is equal to $8(2-\sqrt3)$ for the two dimensional normed linear space whose unit sphere is a dodecahedron.