A counterexample to a strengthening of a question of Milman

Abstract

Let $|\cdot |$ be the standard Euclidean norm on ${\mathbb{R}}^n$ and let $X=({\mathbb{R}}^n,\Vert \cdot \Vert )$ be a normed space. A subspace $Y\subset X$ is \emph{strongly $\alpha$-Euclidean} if there is a constant $t$ such that $t|y|\le \Vert y\Vert \le \alpha t|y|$ for every $y\in Y$, and say that it is \emph{strongly $\alpha$-complemented} if $\Vert P_Y\Vert \le \alpha$, where $P_Y$ is the orthogonal projection from $X$ to $Y$ and $\Vert P_Y\Vert$ 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 $(1+\epsilon )$-Euclidean and strongly $(1+\epsilon )$-complemented, where $\epsilon >0$ is an absolute constant. This example is closely related to an old question of Vitali Milman.

21 pages