| Résume | In this talk I will discuss a version of an old question of Vitali Milman about almost Euclidean and well-complemented subspaces. In particular, I will introduce a notion of `$\epsilon$-good points', which allows for a convenient reformulation of the problem. Let $(X, \|\cdot\|_X)$ be a normed space. It turns out that if a linear subspace $Y\subset X$ consists entirely of $\epsilon$-good points then the restriction of the norm $\|\cdot \|_X$ to $Y$ must be approximately a multiple of the $\ell_2$ norm and the operator norm of the orthogonal projection onto $Y$ is close to 1. I will present an example of a normed space $X$ of arbitrarily high dimension, whose Banach-Mazur distance from the $\ell_2^{{\rm dim} X}$ is at most $2$, but such that non of its (even two-dimensional) subspaces consists entirely of $\epsilon$-good points. The talk is based on joint work with Timothy Gowers. |
