Definition 13.5.4 (Banach-Mazur Distance).label Let $K,L\suf\R^{n}$ be origin-symmetric convex bodies. The Banach-Mazur distance between $K$ and $L$ is
\begin{align*}d_{BM}(K,L)\define \inf \curl{d\geq 1:\exists T\in GL(n,\R),K\suf TL\suf dK}\end{align*}
Equivalently $d_{BM}(K,L)\leq d$ if and only if there is a linear isomorphism $T$ such that
\begin{align*}\frac{1}{d}\norm{x}{K}\leq\norm{T\inv x}{L}\leq\norm{x}{K}&&\fall x\in\R^{n}\end{align*}
Additionally $\norm{T}{}\norm{T\inv}{}\leq d$. We have several properties of $d_{BM}$. Let $K,L,M$ be origin-symmetric convex bodies in $\R^{n}$
- (1)
$d_{BM}(K,L)\geq 1$ with equality if and only if $K$ and $L$ are linearly equivalent, i.e. $L=TK$ for some $T\in GL(n, R)$
- (2)
$d_{BM}(K,L)=d_{BM}(L,K)$
- (3)
$d_{BM}(K,M)\leq d_{BM}(K,L)d_{BM}(L,M)$
Consequently $\log d_{BM}$ defines a metric on the set of origin-symmetric convex bodies in $\R^{n}$ modulo the equivalence relation of being linearly equivalent.
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