Proximity to ℓ1and Distortion in Asymptotic L1Spaces

For an asymptotic ℓ1spaceXwith a basis (xi) certain asymptotic ℓ1constants,δα(X) are defined forα<ω1.δα(X) measures the equivalence between all normalized block bases (yi)ki=1of (xi) which areSα-admissible with respect to (xi) (Sαis theαth-Schreier class of sets) and the unit vector basis of ℓk1. This leads to the concept of the delta spectrum ofX,Δ(X), which reflects the behavior of stabilized limits ofδα(X). The analogues of these constants under all renormings ofXare also defined and studied. We investigateΔ(X) both in general and for spaces of bounded distortion. We also prove several results on distorting the classical Tsirelsonʹs spaceTand its relatives.