## 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 X_{1} = (ℝ^{n}, ||-||_{1}) with the property that if a space X _{2} = (ℝ^{n}, ||-||_{2}) is "not too far" from X_{1} then there exists a [λn]-dimensional subspace E⊂ℝ^{n} such that E_{1} = (E_{2}, ||-||_{2}) and E2 = (E, ||-||_{2}) are "very close." We then show that such an X_{1} 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