Distances between normed spaces, their subspaces and quotient spaces (Q1073296)

Let \(K_ n\) be the family of all n-dimensional normed spaces, equipped with the Banach-Mazur distance \(d(X,Y)=\inf \{\| T\| \cdot \| T^{-1}\|\); T:X\(\to Y\) is a linear isomorphism\(\}\). It is shown in the paper, for example, that \(d(E,E^*)\) for an arbitrary \(E\in K_ n\) is significantly less than the diam \(K_ n\). In another direction, it is shown that for an arbitrary 2n-dimensional space there exists an n-dimensional subspace with a distance to any cotype 2-space at most \(\sqrt{n}\) up to a logarithmic factor. Moreover, for two arbitrary 2n-dimensional spaces E and F one may find an n-dimensional subspace of E and an n-dimensional quotient space of F with the distance between them at most \(\sqrt{n}\) (up to, again, a logarithmic factor).