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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-3-1,
author = {Alexander E. Litvak and Vitali D. Milman and Nicole Tomczak-Jaegermann},
title = {Essentially-Euclidean convex bodies},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {207-221},
zbl = {1200.46019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-3-1}
}
Alexander E. Litvak; Vitali D. Milman; Nicole Tomczak-Jaegermann. Essentially-Euclidean convex bodies. Studia Mathematica, Tome 196 (2010) pp. 207-221. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-3-1/