Розмернось и мягкие пучки

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 $i n f_{e l l i p s o i d ε \subset B_{E}} {(v o l (B_{E}) / v o l (ε))}^{1 / n} \leq c_{0} i n f_{z o n o i d Z \subset B_{F}} {(v o l (B_{F}) / v o l (Z))}^{1 / k}$ . The concept of volume ratio with respect to $ℓ_{p}$ -spaces is used to prove the following distance estimate for $2 \leq q \leq p < \infty$ : $s u p_{F \subset ℓ_{p}, d i m F = n} i n f_{G \subset L_{q}, d i m G = n} d (F, G) \sim_{c_{p q}} n^{(q / 2) (1 / q - 1 / p)}$ .

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B. Diviš, B. Novák (1974)

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