Volume ratios in L p -spaces

Yehoram Gordon; Marius Junge

Studia Mathematica (1999)

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

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 i n f e l l i p s o i d ε ⊂ B E ( v o l ( B E ) / v o l ( ε ) ) 1 / n ≤ c 0 i n f z o n o i d Z ⊂ 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 ≤ q ≤ p < ∞ : s u p F ⊂ ℓ p , d i m F = n i n f G ⊂ L q , d i m G = n d ( F , G ) ∼ c p q n ( q / 2 ) ( 1 / q - 1 / p ) .

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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