Essentially-Euclidean convex bodies

• Autorzy: Alexander E. Litvak , Vitali D. Milman , Nicole Tomczak-Jaegermann
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 52A23: Asymptotic theory of convex bodies
• 46B20: Geometry and structure of normed linear spaces
• 46B06: Asymptotic theory of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 196
• Numer: 3
• Strony: 207-221

Daty

wydano

2010

Twórcy

autor

Alexander E. Litvak

• Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

autor

Vitali D. Milman

• Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

autor

Nicole Tomczak-Jaegermann

• Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Identyfikatory

DOI

10.4064/sm196-3-1