Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T04:53:15Z http://mathoverflow.net/feeds/question/22028 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22028/diameter-of-a-metric-on-orbits-under-affine-bijections-of-n-dimensional-convex Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets Roland Bacher 2010-04-21T08:16:48Z 2010-04-21T13:42:15Z <p>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.</p> <p>The function $d$ defines thus a distance on the set $\mathcal C_n$ of orbits under affine bijections of $n-$dimensional convex compact sets.</p> <p>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.</p> http://mathoverflow.net/questions/22028/diameter-of-a-metric-on-orbits-under-affine-bijections-of-n-dimensional-convex/22042#22042 Answer by Mark Meckes for Diameter of a metric on orbits under affine bijections of $n-$dimensional convex compact sets Mark Meckes 2010-04-21T11:45:26Z 2010-04-21T13:24:23Z <p>I assume you also want your compact sets to have non-empty interior, hence positive volume.</p> <p>The literature mostly deals with the related <a href="http://en.wikipedia.org/wiki/Banach-Mazur_compactum" rel="nofollow">Banach-Mazur</a> 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:</p> <p>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 <code>$(\mathcal{C}_n, d_{BM})$</code> is at most $2\log n$. I believe the exact diameter is an open question.</p> <p>Let $\mathcal{C}_n^0$ be the set of affine equivalence classes of centrally symmetric convex bodies. Then if $B$ is a Euclidean ball, <code>$d_{BM}(A,B) \le \log \sqrt{n}$</code>, with equality achieved when $A$ is a cube or a crosspolytope (but not only then); therefore the diameter of <code>$\mathcal{C}_n^0,d_{BM})$</code> is at most $2\log\sqrt{n} = \log n$. Gluskin proved that the diameter of <code>$(\mathcal{C}_n^0,d_{BM})$</code> 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 <code>$(\mathcal{C}_n^0,d)$</code>.</p>