Let $K \subset \mathbb{R}^n$ be an isotropic convex body (i.e : barycenter at the origin, volume 1, covariance matrix a multiple of the identity).
The question of the greatest euclidean ball inscribed in K, meaning the greatest $c>0$ such that $cB_2^n \subset K$, where $B_2^n$ is often asked, but I would like to know if there is something known about the greatest hypercube inside $K$ ?
Can we hope for something like : There exists $c>0$ such that for every n, for every isotropic convex body $K$ of $\mathbb{R}^n$, there exists a rotation $R \in \mathcal{O}(n)$ such that :
$$ cRB_{\infty}^n \subset K$$
Where $B_{\infty}^n$ is the hypercube of dimension n
In other words, setting $$c(K) = \sup\{c>0 \ \text{such that} \ \exists R\in \mathcal{O}(n), \ cRB_{\infty}^n \subset K \} $$
and
$$c_n = \inf\{c(K), \ K \ \text{isotropic convex body of } \mathbb{R}^n\} $$
I'm looking for lower bound on $c_n$ and hoping that $c_n \geq c$ for some absolute constant $c$. Of course one has the trivial upper-bound $c_n \leq 2$ for volume reasons