Some results on the Dunkl-Williams constant

This paper presents a compilation of various formulas for calculating the Dunkl-Williams constant $DW(X)$ of a real normed linear space. The constant $D{W}_B(X)$ 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 ${\mathrm{\ell }}_2-{\mathrm{\ell }}_1$ is equal to $2\sqrt{2}$, and that it is equal to $8(2-\sqrt{3})$ for the two dimensional normed linear space whose unit sphere is a dodecahedron.