= Banach-Mazur distance
{c}
{title2=$d(E,F)=\inf_S\|S\|\|S^{-1}\|$}
For isomorphic finite-dimensional <normed vector spaces>, the Banach-Mazur distance is the infimum of $\|S\|\|S^{-1}\|$ over their linear <isomorphisms>. If an $N$-dimensional subspace $E$ of a <Banach space> of <cotype 2> satisfies the summing estimate $\pi_2(T)\le L\|T\|$ for operators from $\ell_\infty^N$, then $d(E,\ell_\infty^N)\ge\sqrt N/L$: compose an isomorphism with the inclusion and use $\pi_2(I_E)=\sqrt N$.
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