# Volume ratios in ${L}_{p}$-spaces

Studia Mathematica (1999)

• Volume: 136, Issue: 2, page 147-182
• ISSN: 0039-3223

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## Abstract

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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 $in{f}_{ellipsoid\epsilon \subset {B}_{E}}{\left(vol\left({B}_{E}\right)/vol\left(\epsilon \right)\right)}^{1/n}\le {c}_{0}in{f}_{zonoidZ\subset {B}_{F}}{\left(vol\left({B}_{F}\right)/vol\left(Z\right)\right)}^{1/k}$ . The concept of volume ratio with respect to ${\ell }_{p}$-spaces is used to prove the following distance estimate for $2\le q\le p<\infty$: $su{p}_{F\subset {\ell }_{p},dimF=n}in{f}_{G\subset {L}_{q},dimG=n}d\left(F,G\right){\sim }_{{c}_{pq}}{n}^{\left(q/2\right)\left(1/q-1/p\right)}$.

## How to cite

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Gordon, Yehoram, and Junge, Marius. "Volume ratios in $L_p$-spaces." Studia Mathematica 136.2 (1999): 147-182. <http://eudml.org/doc/216665>.

@article{Gordon1999,
abstract = {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 $2≤ q≤ p < ∞$: $sup_\{F ⊂ ℓ_p, dim F=n\} inf_\{G ⊂ L_q, dim G=n\} d(F,G) ∼_\{c_\{pq\}\} n^\{(q/2)(1/q-1/p)\}$.},
author = {Gordon, Yehoram, Junge, Marius},
journal = {Studia Mathematica},
keywords = {Carl's inequality; Chevet's inequality; ellipsoid; entropy numbers; Gaussian variables; Gelfand numbers; Gluskin's inequality; volume ratio with respect to ellipsoids; Gordon-Lewis constant; inverse Santaló’s inequality; -convexity; space; space; Marcus-Pisier's theorem; ideal spaces of operators; Gordon-Lewis norms; Meyer-Pajor's inequality; Pajor-Tomczak's inequality; -summing operator; random operator; Santaló’s inequality; volume ratio numbers; zonoid},
language = {eng},
number = {2},
pages = {147-182},
title = {Volume ratios in $L_p$-spaces},
url = {http://eudml.org/doc/216665},
volume = {136},
year = {1999},
}

TY - JOUR
AU - Gordon, Yehoram
AU - Junge, Marius
TI - Volume ratios in $L_p$-spaces
JO - Studia Mathematica
PY - 1999
VL - 136
IS - 2
SP - 147
EP - 182
AB - 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 $2≤ q≤ p < ∞$: $sup_{F ⊂ ℓ_p, dim F=n} inf_{G ⊂ L_q, dim G=n} d(F,G) ∼_{c_{pq}} n^{(q/2)(1/q-1/p)}$.
LA - eng
KW - Carl's inequality; Chevet's inequality; ellipsoid; entropy numbers; Gaussian variables; Gelfand numbers; Gluskin's inequality; volume ratio with respect to ellipsoids; Gordon-Lewis constant; inverse Santaló’s inequality; -convexity; space; space; Marcus-Pisier's theorem; ideal spaces of operators; Gordon-Lewis norms; Meyer-Pajor's inequality; Pajor-Tomczak's inequality; -summing operator; random operator; Santaló’s inequality; volume ratio numbers; zonoid
UR - http://eudml.org/doc/216665
ER -

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