Suppose we have a convex compact body $D\subset \mathbb R^n$. We can try to apply affine transformation keeping the volume and decreasing the diameter of $D$.

It is clear that there is a constant $\lambda_n$ such that for any $D$ there is an affinity $F$ such that $diamater(F(D))^n\leq\lambda_n Volume(F(D))$. I'm interesting in the optimal value for $\lambda_n$.

The article "On the thinnest non-separable lattice of convex bodies" (E. Makai, p.23) gives an estimate $\lambda_n\leq (_n^{2n})n^{n/2}/k_n$ where $k_n$ is the volume of the convex hull of the unit sphere and $(\pm\sqrt{n},0,0,..)$, $\lambda_2$ is also computed.

I can not find in literature any better estimate for $\lambda_3$, but it seems that I can prove a few better estimate by school-like methods: if there is no affinity decreasing diameter, that means that in $D$ there are a lot of diameters which will increase if we apply an infinitesimal affinity, so we get an estimate. I'm sure that I'm not the first one who apply such a simple idea to this problem. Do you know any others references concerning this problem?