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?



On the dependence on $\epsilon$ in Dvoretzky's theorem
Schechtmann replaced the $\varepsilon^2$ in Gordon's bound by $\varepsilon$


Two observations regarding embedding subsets of Euclidean spaces in normed spaces, Advances in Mathematics 200(1), 125-135 (2006), [doi:10.1016/j.aim.2004.11.003](http://dx.doi.org/10.1016/j.aim.2004.11.003) 


Euclidean Sections of Convex Bodies, In Asymptotic Geometric Analysis, Volume 68 of the series Fields Institute Communications, 271-288 (2013), [doi:10.1007/978-1-4614-6406-8\_12](http://dx.doi.org/10.1007/978-1-4614-6406-8_12), [arxiv:1110.6401](http://arxiv.org/abs/1110.6401)


