Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm.

**Question:** Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so that the Lipschitz constant of $f_N$ is $C$. Is $C_N$ bounded as $N\to \infty$?

Notice that the $\ell_\infty^N$ norm is $2$-equivalent to the $\ell_p^N$ norm with $p = \log_2 N$, and the Mazur map from $B_p^N$ to $B_2^N$ has Lipschitz constant about $p$, so $C_N$ grows no faster than $\log N$.

(The Mazur map from $\ell_p$ to $\ell_2$ is defined for unit vectors to be $x\mapsto |x|^{p/2}\rm{sign}\ x$ and extended to be positively homogeneous on $\ell_p$.)