A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

We prove that if \(X\) is an \(\ell_{1}\)-predual isomorphic to the space \(c_{0}\) of sequences converging zero, then for any isomorphism \(T:X\rightarrow c_{0}\) we have \(\Vert T\Vert\, \Vert T^{-1}\Vert\ge1+2r^{*}(X)\), where \(r^{*}(X)\) smallest radius closed ball dual \(X^{*}\) containing all weak\(^{*}\) cluster points set extreme unit \(X^*\).