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 $\lambda$-essentially-Euclidean (with $0 < \lambda <1$) if it has a $[\lambda n]$-dimensional subspace which has further proportional-dimensional Euclidean subspaces of any proportion. We consider a space $X_1=(\mathbb R^n, \|\cdot \|_1)$ with the property that if a space $X_2=(\mathbb R^n, \|\cdot \|_2)$ is “not too far” from $X_1$ then there exists a $[\lambda n]$-dimensional subspace $E\subset \mathbb R^n$ such that $E_1=(E, \|\cdot \|_1)$ and $E_2=(E, \|\cdot \|_2)$ are “very close.” We then show that such an $X_1$ is $\lambda$-essentially-Euclidean (with $\lambda$ 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.