Essentially-Euclidean convex bodies

Alexander E. Litvak, Vitali D. Milman, Tomczak Jaegermann Nicole

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)207-221
Number of pages15
JournalStudia Mathematica
Volume196
Issue number3
DOIs
StatePublished - 2010

Keywords

  • Banach - Mazur distance
  • Convex bodies
  • Covering numbers
  • Essentially euclidean convex bodies
  • Euclidean sections
  • Geometric distance
  • M - Ellipsoid
  • M - Position
  • Projections

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