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 proportionaldimensional Euclidean subspaces of any proportion. We consider a space X1 = (ℝn, ||-||1) with the property that if a space X 2 = (ℝn, ||-||2) is "not too far" from X1 then there exists a [λn]-dimensional subspace E⊂ℝn such that E1 = (E2, ||-||2) 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.
Original language | English |
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Pages (from-to) | 207-221 |
Number of pages | 15 |
Journal | Studia Mathematica |
Volume | 196 |
Issue number | 3 |
DOIs | |
State | Published - 2010 |
Keywords
- Banach - Mazur distance
- Convex bodies
- Covering numbers
- Essentially euclidean convex bodies
- Euclidean sections
- Geometric distance
- M - Ellipsoid
- M - Position
- Projections