A note on the Banach–Mazur distances between $c_0$ and other $\ell_1$-preduals

Abstract

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