 Volume ratio

The volume ratio of a $d$-dimensional normed vector space is $\mathrm{vr}(E)=(\mathrm{vol}(B_E)/\mathrm{vol}(\mathcal{E}){)}^{1/d}$, where $\mathcal{E}$ is its John ellipsoid. It is unchanged by invertible linear changes of coordinates. In John position, polar integration gives $\mathrm{vr}(E{)}^d={\int }_{S^{d-1}}\Vert u{\Vert }_E^{-d}d\sigma (u)$, with normalized spherical measure. Consequently $\sigma \{\Vert u{\Vert }_E\le t\}\le (\mathrm{vr}(E)t{)}^d$.