Essentially-Euclidean convex bodies

Alexander E. Litvak, Vitali D. Milman, and Nicole Tomczak-Jaegermann. "Essentially-Euclidean convex bodies." Studia Mathematica 196.3 (2010): 207-221. <http://eudml.org/doc/285418>.

@article{AlexanderE2010,
abstract = {In this note we introduce a notion of essentially-Euclidean normed spaces (and convex bodies). Roughly speaking, an n-dimensional space is λ-essentially-Euclidean (with 0 < λ < 1) if it has a [λn]-dimensional subspace which has further proportional-dimensional Euclidean subspaces of any proportion. We consider a space X₁ = (ℝⁿ,||·||₁) with the property that if a space X₂ = (ℝⁿ,||·||₂) is "not too far" from X₁ then there exists a [λn]-dimensional subspace E⊂ ℝⁿ such that E₁ = (E,||·||₁) and E₂ = (E,||·||₂) are "very close." We then show that such an X₁ is λ-essentially-Euclidean (with λ depending only on quantitative parameters measuring "closeness" of two normed spaces). This gives a very strong negative answer to an old question of the second named author. It also clarifies a previously obtained answer by Bourgain and Tzafriri. We prove a number of other results of a similar nature. Our work shows that, in a sense, most constructions of the asymptotic theory of normed spaces cannot be extended beyond essentially-Euclidean spaces.},
author = {Alexander E. Litvak, Vitali D. Milman, Nicole Tomczak-Jaegermann},
journal = {Studia Mathematica},
keywords = {Milman ellipsoid; finite volume ratio; Dvoretzky's theorem; convex body},
language = {eng},
number = {3},
pages = {207-221},
title = {Essentially-Euclidean convex bodies},
url = {http://eudml.org/doc/285418},
volume = {196},
year = {2010},
}

TY - JOUR
AU - Alexander E. Litvak
AU - Vitali D. Milman
AU - Nicole Tomczak-Jaegermann
TI - Essentially-Euclidean convex bodies
JO - Studia Mathematica
PY - 2010
VL - 196
IS - 3
SP - 207
EP - 221
AB - In this note we introduce a notion of essentially-Euclidean normed spaces (and convex bodies). Roughly speaking, an n-dimensional space is λ-essentially-Euclidean (with 0 < λ < 1) if it has a [λn]-dimensional subspace which has further proportional-dimensional Euclidean subspaces of any proportion. We consider a space X₁ = (ℝⁿ,||·||₁) with the property that if a space X₂ = (ℝⁿ,||·||₂) is "not too far" from X₁ then there exists a [λn]-dimensional subspace E⊂ ℝⁿ such that E₁ = (E,||·||₁) and E₂ = (E,||·||₂) are "very close." We then show that such an X₁ is λ-essentially-Euclidean (with λ depending only on quantitative parameters measuring "closeness" of two normed spaces). This gives a very strong negative answer to an old question of the second named author. It also clarifies a previously obtained answer by Bourgain and Tzafriri. We prove a number of other results of a similar nature. Our work shows that, in a sense, most constructions of the asymptotic theory of normed spaces cannot be extended beyond essentially-Euclidean spaces.
LA - eng
KW - Milman ellipsoid; finite volume ratio; Dvoretzky's theorem; convex body
UR - http://eudml.org/doc/285418
ER -