Essentially-Euclidean convex bodies

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 proportionally 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 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.