On certain equivalent norms on Tsirelson's space

Abstract

Tsirelson's space $T$ is known to be distortable but it is open as to whether or not $T$ is arbitrarily distortable. For $n \in \mathbb{N}$ the norm $||\cdot||_{n}$ of the Tsirelson space $T(S_{n},2^{-n})$ is equivalent to the standard norm on $T$. We prove there exists $K \lt \infty$ so that for all $n$, ||\cdot||_{n} does not $K$ distort any subspace $Y$ of $T$.