A $k$-dimensional section of a convex body $K \subset {\mathbb R}^n$ is just the intersection of $K$ with a $k$-dimensional hyperplane $h$.

Such a section is said to be $(1+\epsilon )$-almost spherical if $B(0,\frac {R}{1 + \epsilon }) \subset h \cap K \subset B(0, (1 + \epsilon )R)$, where $B(0,R)$ denote the Euclidean ball of radius $R$ about the origin.

Dvoretzky's theorem states that any centrally symmetric convex body $K \subset {\mathbb R}^n$, with non-empty interior contains a $k$-dimensional section which is $(1 + \epsilon )$-almost spherical, provided $n \geq n_0(k,\epsilon )$.

My question is about the function $n_0(k,\epsilon )$. Milman proved that $n_0(k,\epsilon ) \leq \epsilon ^{-ck\epsilon ^2}$, for some constant $c>0$. Gordon later improved this to $n_0(k,\epsilon )\leq 2^{ck/\epsilon ^2}$. Is this dependence on $\epsilon $ tight?