A counterexample to a strengthening of a question of V. D. Milman

Gowers, Timothy; Wyczesany, Katarzyna

Annales Henri Lebesgue, Tome 6 (2023), pp. 427-448

Let $|·|$ be the standard Euclidean norm on ${ℝ}^n$ and let $X=({ℝ}^n,\parallel ·\parallel )$ be a normed space. A subspace $Y\subset X$ is strongly $\alpha$-Euclidean if there is a constant $t$ such that $t\left|y\right|\le \parallel y\parallel \le \alpha t\left|y\right|$ for every $y\in Y$, and say that it is strongly $\alpha$-complemented if $\parallel P_Y\parallel \le \alpha$, where $P_Y$ is the orthogonal projection from $X$ to $Y$ and $\parallel P_Y\parallel$ denotes the operator norm of $P_Y$ with respect to the norm on $X$. We give an example of a normed space $X$ of arbitrarily high dimension that is strongly 2-Euclidean but contains no 2-dimensional subspace that is both strongly $(1+ϵ)$-Euclidean and strongly $(1+ϵ)$-complemented, where $ϵ>0$ is an absolute constant. This property is closely related to an old question of Vitali Milman. The example is probabilistic in nature.

Soit $|·|$ la norme euclidienne standard sur ${ℝ}^n$ et soit $X=({ℝ}^n,\parallel ·\parallel )$ un espace normé. Un sous-espace $Y\subset X$ est fortement $\alpha$-euclidien s’il existe une constante $t$ telle que $t\left|y\right|\le \parallel y\parallel \le \alpha t\left|y\right|$ pour tout $y\in Y$, et fortement $\alpha$-complémenté si $\parallel P_Y\parallel \le \alpha$, où $P_Y$ est la projection orthogonale de $X$ sur $Y$ et $\parallel P_Y\parallel$ désigne la norme d’opérateur de $P_Y$ par rapport à la norme sur $X$. Nous donnons un exemple d’un espace normé $X$ avec une dimension arbitrairement grande qui est fortement $2$-euclidien mais ne contient pas de sous-espace à $2$ dimensions qui soit à la fois fortement $(1+ϵ)$-euclidien et fortement $(1+ϵ)$-complémenté, où $ϵ>0$ est une constante absolue. Cette propriété est étroitement liée à une vieille question de Vitali Milman. L’exemple est de nature probabiliste.

Classification : 46B06, 52A23, 46B09, 46B20
Keywords: normed space, almost Euclidean, well complemented

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