Given Banach spaces $X$ and $Y$, the Banach-Mazur distance between $X$ and $Y$ is defined as $$ d(X,Y) = \inf\{ \|\varphi\|\|\varphi^{-1}\| : \varphi\colon X\to Y \text{ isomorphism} \}. $$ We consider the Banach spaces $\ell_\infty=\ell_\infty(\mathbb{N})$ and $L_\infty=L_\infty([0,1],\lambda)$, where $\lambda$ is the Lebesgue measure on $[0,1]$.

What is $d(\ell_\infty,L_\infty)$?