The literature mostly deals with the related Banach-Mazur metric $d_{BM}(A,B)$, in which it is assumed that $\alpha_1(A)$ and $\alpha_2(B)$ \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$\log \sqrt{n} + n - c$for a constant$c$independent of$n$, by in fact proving the same lower bound for the diameter of$(\mathcal{C}_n^0,d)$. 1 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$\alpha_2(B)$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$\log \sqrt{n} + c$for a constant$c$independent of$n$, by in fact proving the same lower bound for the diameter of$(\mathcal{C}_n^0,d)\$.