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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.

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Added the banach-spaces tag, even though no Banach spaces appear explicitly in the question, because Banach space specialists are likely to have the most information about it. –  Mark Meckes Apr 21 '10 at 13:43
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1 Answer 1

up vote 3 down vote accepted

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 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 $(\mathcal{C}_n^0,d)$.

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Thank you for these precisions. The two metrics are however of a different nature since the underlying sets are different: The Banach-Mazur metric is defined on the set of convex compact sets while the above metric is defined for orbits under affine bijections of convex compact sets. By the way, one of my questions can be easily reformulated for the Banach-Mazur metric: Is it true that the $n-$dimensional simplex is at maximal distance (measured by $d_{BM}$) from the $n-$dimensional ball? –  Roland Bacher Apr 21 '10 at 15:06
    
The Banach-Mazur distance is also defined on the set of affine equivalence classes. A cousin of it, defined on convex bodies themselves without minimizing over affine transformations, is sometimes called "geometric distance". The answer to your last question, as I indicated in the answer above, is yes. –  Mark Meckes Apr 21 '10 at 15:13
    
You are of course right. –  Roland Bacher Apr 22 '10 at 16:09
    
I accept this answer although it does not answer my question completely, the diameter of these metric spaces is still unknown to me. By the way, this question is related to another question (more interesting, in my eyes) on which I am also completely clueless: Can a discrete set of the plane of uniform density intersect all large triangles? –  Roland Bacher Apr 26 '10 at 14:34
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