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 $n\in {\Bbb N}$ 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 $K<\infty$ so that for all $n$, $\|\cdot\|_n$ does not $K$ distort any subspace $Y$ of $T$.