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# Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets

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.