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Volume ratios in $L_p$-spaces

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There exists an absolute constant $c_0$ such that for any n-dimensional Banach space E there exists a k-dimensional subspace F ⊂ E with k≤ n/2 such that $inf_{ellipsoid ε ⊂ B_E} (vol(B_E)/vol(ε))^{1/n} ≤ c_0 inf_{zonoid Z ⊂ B_F} (vol(B_F)/vol(Z))^{1/k}$ . The concept of volume ratio with respect to $ℓ_p$-spaces is used to prove the following distance estimate for $2≤ q≤ p < ∞$: $sup_{F ⊂ ℓ_p, dim F=n} inf_{G ⊂ L_q, dim G=n} d(F,G) ∼_{c_{pq}} n^{(q/2)(1/q-1/p)}$.
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