Given two $n-$dimensional convex compact sets $A,B$, we define $d(A,B)$ as $\log({\mathrm{Vol}}(\alpha\_2(A)))-\log(\mathrm{Vol}(\alpha\_1(A)))$ where $\alpha\_1,\alpha\_2$ are two affine bijections such that $\alpha\_1(A)\subset B\subset\alpha\_2(A)$ and such that the ratio $\mathrm{Vol}(\alpha\_2(A))/\mathrm{Vol}(\alpha\_1(A))$ is minimal. The function 
$d$ is symmetric, satisfies the triangle inequality, is well-defined for orbits of
convex sets under affine bijections and $d(A,B)=0$ if and only if $A$ and $B$ are in the same orbit under affine bijections.


The function $d$ defines thus a distance on the set $\mathcal C\_n$ of orbits under affine bijections of $n-$dimensional convex compact sets.


What is the diameter of the metric set $\mathcal C\_n$?
(It is easy to see that $\mathcal C\_n$ is of bounded diameter.) A natural guess is that 
the diameter is achieved by the distance of (the orbit of) an $n-$dimensional ball to
(the orbit of) the $n-$dimensional simplex.



Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets
I assume you also want your compact sets to have non-empty interior, hence positive volume.


The literature mostly deals with the related [Banach-Mazur](http://en.wikipedia.org/wiki/Banach-Mazur_compactum) metric $d\_{BM}(A,B)$, in which it is assumed that $\alpha\_1(A)$ and $\alpha\_2(A)$ are homothetic, so $d\_{BM}(A,B) \ge d(A,B)$. (Here I'm following your convention and making $d\_{BM}$ a metric, as opposed to the usual definition which makes its logarithm a metric.) Here's a little of what's known about that related to your question:


If $B$ is a Euclidean ball, then $d\_{BM}(A,B) \le \log n$, with equality achieved exactly when $A$ is a simplex. Thus the diameter of $(\mathcal{C}\_n, d\_{BM})$ is at most $2\log n$. I believe the exact diameter is an open question.


Let $\mathcal{C}\_n^0$ be the set of affine equivalence classes of centrally symmetric convex bodies. Then if $B$ is a Euclidean ball, $d\_{BM}(A,B) \le \log \sqrt{n}$, with equality achieved when $A$ is a cube or a crosspolytope (but not only then); therefore the diameter of $\mathcal{C}\_n^0,d\_{BM})$ is at most $2\log\sqrt{n} = \log n$. Gluskin proved that the diameter of $(\mathcal{C}\_n^0,d\_{BM})$ is at least $\log n - c$ for a constant $c$ independent of $n$, by in fact proving the same lower bound for the diameter of $(\mathcal{C}\_n^0,d)$.


