= Volume ratio
{title2=$\operatorname{vr}(E)$}
The <volume ratio> of a $d$-dimensional <normed vector space> is $\operatorname{vr}(E)=(\operatorname{vol}(B_E)/\operatorname{vol}(\mathcal E))^{1/d}$, where $\mathcal E$ is its <John ellipsoid>. It is unchanged by invertible linear changes of coordinates. In <John position>, polar integration gives $\operatorname{vr}(E)^d=\int_{S^{d-1}}\|u\|_E^{-d}\,d\sigma(u)$, with normalized spherical measure. Consequently $\sigma\{\|u\|_E\leq t\}\leq(\operatorname{vr}(E)t)^d$.
Back to article page
