= Orthogonal splitting with bounded volume ratio
{title2=$L\leq32\operatorname{vr}(E)^2$}
In a $2k$-dimensional <normed vector space> in <John position>, two orthogonal $k$-dimensional subspaces admit $\|x\|_E\leq |x|\leq32\operatorname{vr}(E)^2\|x\|_E$. To see this, take $\varepsilon$-nets in two orthogonal half-dimensional unit spheres, each of size at most $(1+2/\varepsilon)^k$, and rotate them uniformly together. The <volume ratio> bound makes the <probability> that any net point has <norm> below $2\varepsilon$ at most $2[4\operatorname{vr}(E)^2(\varepsilon^2+2\varepsilon)]^k$. With $\varepsilon=(32\operatorname{vr}(E)^2)^{-1}$ this is less than $1$. Since $\|\cdot\|_E$ is $1$-<Lipschitz> for the <Euclidean metric>, the successful nets give the stated inequality everywhere on both spheres. A dimension-independent constant therefore requires a bound on the <volume ratio>; it does not hold for arbitrary <normed vector spaces>.
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