Essentially-Euclidean convex bodies

Alexander E. Litvak¹ ; Vitali D. Milman² ; Nicole Tomczak-Jaegermann¹

¹Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2G1
²Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel

Studia Mathematica, Tome 196 (2010) no. 3, pp. 207-221

Résumé

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.

Détail

DOI : 10.4064/sm196-3-1

Keywords: note introduce notion essentially euclidean normed spaces convex bodies roughly speaking n dimensional space lambda essentially euclidean lambda has lambda dimensional subspace which has further proportional dimensional euclidean subspaces proportion consider space mathbb cdot property space mathbb cdot too far there exists lambda dimensional subspace subset mathbb cdot cdot close lambda essentially euclidean lambda depending only quantitative parameters measuring closeness normed spaces gives strong negative answer old question second named author clarifies previously obtained answer bourgain tzafriri prove number other results similar nature work shows sense constructions asymptotic theory normed spaces cannot extended beyond essentially euclidean spaces