Banach–Mazur distance

Summary

The Banach–Mazur distance measures how far two normed vector spaces of the same finite dimension are from being isometric. It is the smallest possible distortion $\norm{T} \norm{T^{-1}}$ of a linear isomorphism $T$ between them. In geometric terms, it compares the unit balls up to linear maps. With it, the isometry classes of $n$-dimensional normed spaces form the Banach–Mazur compactum. Many questions of local Banach space theory and asymptotic convex geometry are estimates of this distance.