A counterexample to a strengthening of a question of Milman

Abstract: Let $|\cdot|$ be the standard Euclidean norm on $\mathbb{R}^n$ and let $X=(\mathbb{R}^n,|\cdot|)$ be a normed space. A subspace $Y\subset X$ is \emph{strongly $\alpha$-Euclidean} if there is a constant $t$ such that $t|y|\leq|y|\leq\alpha t|y|$ for every $y\in Y$, and say that it is \emph{strongly $\alpha$-complemented} if $|P_Y|\leq\alpha$, where $P_Y$ is the orthogonal projection from $X$ to $Y$ and $|P_Y|$ 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.