Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is: 


**1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" isomorphisms between these two spaces?**


Since $L^\infty$ is 1-injective, we have the lower bound 
$$
inf\{\|P\|: \mbox{$P$ is a projection from $L^\infty$ onto $H$}\}\leq d\_{BM}(H,L^\infty). 
$$
Recalling that the canonical projection $P:L^\infty\rightarrow H$ given by $Pf(x)= f(x) - \int f$ has norm $2$, we can pose the following related question: 


**2. Given a (linear, continuous) projection $P:L^\infty\rightarrow H$, do we always have $\|P\|\geq 2$?**



On hyperplanes of $L\infty$
IIRC, if $K$ is a compact Hausdorff space that has no isolated points, then every projection from $C(K)$ onto a hyperplane has norm at least two. Maybe Dan Amir proved this? Anyway, in your situation the proof goes like this. A projection $P$ from $L^\infty$ onto $H$ has the form $Pf = f -(\int f) g$, where $\int g = 1$. Let $E=[g\ge 1]$ and observe that $g$ has positive measure. Take a subset $F$ of $E$ that has measure $\epsilon > 0$ and let $h=1\_F - 1\_{F^c}$. Then $H$ has norm one in $L^\infty$ and $\int h = -1 + 2\epsilon$. Thus $Ph$ is at least $2 - 2\epsilon$ on $F$.


