Let $K,L\in\mathcal{K}^n$ be two convex bodies, their Banach-Mazur distance is defined as
$$d_{BM}(K,L)=\inf\{\lambda>0\colon\exists x,y\in\mathbb{R}^n,T\in GL_n(\mathbb{R})/y+K\subset T(x+L)\subset\lambda(y+K)\}.$$
Where $GL_n(\mathbb{R}^n)$ is the set of linear transformations. It's easy to see that the distance is invariant by affine transformations, and that is verifies the following properties:
$d(K,x+TK)=1$ (reflexivity-like).
$d(K,L)=d(L,K)$ (symmetry).
$d(K,M)\leq d(K,L)d(L,M)$ (multiplicative triangular inequality).
Which leads to believe that $\log(d_{BM})$ is a distance in the set of convex bodies, up to affine transformations. I've seen this stated, for example in
Lassak, Marek, Banach-Mazur distance of planar convex bodies, Aequationes Math. 74, No. 3, 282-286 (2007). ZBL1137.52002.
page 1 second paragraph, available in
https://link.springer.com/article/10.1007/s00010-007-2874-x
even without mentioning the necessary condition that it can only be up to affine transformations, since this function can't distinguish between two sets that are an affine transformation appart. But I haven't seen the proof.
The only thing, I believe, that remains to prove is that is $\log(d_{BM}(K,L))=1$ then $K,L$ are an affine transformation appart, that is, there exists $T\in GL_n(\mathbb{R}^n)$ and $x\in\mathbb{R}^n$ such that $L=x+TK$.