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## Essentially-Euclidean convex bodies

### Volume 196 / 2010

Studia Mathematica 196 (2010), 207-221 MSC: 46B06, 52A23, 46B20. DOI: 10.4064/sm196-3-1

#### Abstract

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.

#### Authors

• Alexander E. LitvakDepartment of Mathematical
and Statistical Sciences
University of Alberta
e-mail
• Vitali D. MilmanDepartment of Mathematics
Raymond and Beverly Sackler Faculty
of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
e-mail
• Nicole Tomczak-JaegermannDepartment of Mathematical
and Statistical Sciences
University of Alberta