Bounds for projection constants and $1$-summing norms

Abstract:

It is shown that projection constants $\lambda ({X_n})$ of $n$-dimensional normed spaces ${X_n}$ satisfy $\lambda ({X_n}) \leqslant \sqrt n - c/\sqrt n$ where $c > 0$ is a numerical constant. Similarly, the $1$-summing norms of (the identity of) ${X_n}$ can be estimated by ${\pi _1}({X_n}) \geqslant \sqrt n + c/\sqrt n$. These estimates are the best possible: for prime $n$, translation-invariant $n$-dimensional spaces ${X_n}$ such that $\lambda ({X_n}) \geqslant \sqrt n - 2/\sqrt n$ and ${\pi _1}({X_n}) \leqslant \sqrt n + 2/\sqrt n$ can be constructed. For these spaces Gordon-Lewis constants and distances to Hilbert spaces are large as well: $\operatorname {gl} ({X_n}) \geqslant \tfrac {1} {3}\sqrt n ,d({X_n},l_2^n) = \sqrt n$.