TITLE: What is the Banach-Mazur distance between $\ell_\infty$ and $L_\infty$?
QUESTION [28 upvotes]: 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 $d(\ell_\infty,L_\infty)$?

REPLY [2 votes]: The distance is strictly greater than $2$ --- as is true for any compact, nonhomeomorphic $K_1$ and $K_2$ when $K_1$ is totally disconnected (Cohen-Chu, Stud. Math. 1995, p.6).  The question of an upper bound is more puzzling.  @HannesThiel comment 6/13/2016 suggests distance $\le 16$.