Let $X$ be a normed space. I'd like to show that all closed hyperplanes in $X$ are isomorphic.
My attempt
Let $H$ and $W$ be closed hyperplanes. We know $\dim(X/H)=\dim(X/W)=1$, therefore $(X/H)$ and $(X/W)$ are isomorphic. How to conclude that W and H are isomorphic?
Pick $v^*\in X^*$ with $Ker(v^*)=V$. Then, if $v^*(v)\not=0$, $V$ is isomorphic to $X/[v]$. Namely, the maps $$ x\mapsto x+[v], \quad x+[v] \mapsto x-\frac{v^*(x)}{v^*(v)}v $$ are inverse isomorphism.
Pick $w^*$ and $w$ as above corresponding to $W$. There is $u^*\in X^*$ with $u^*(v)$ and $u^*(w)$ nonnull scalars. If $U=Ker(u^*)$ we have $$ V\simeq X/[v]\simeq U \simeq X/[w]\simeq W. $$
A quantitative study reveals that the Banach-Mazur distance from U to W is bounded by a universal constant (not depending of the Banach space $X$). Is the optimal constant known?
