Proximity to ℓ1and Distortion in Asymptotic L1Spaces

Abstract For an asymptotic l 1 space X with a basis ( x i ) certain asymptotic l 1 constants, δ α ( X ) are defined for α ω 1 . δ α ( X ) measures the equivalence between all normalized block bases ( y i ) k i =1 of ( x i ) which are S α -admissible with respect to ( x i ) ( S α is the α th-Schreier class of sets) and the unit vector basis of l k 1 . This leads to the concept of the delta spectrum of X , Δ ( X ), which reflects the behavior of stabilized limits of δ α ( X ). The analogues of these constants under all renormings of X are also defined and studied. We investigate Δ ( X ) both in general and for spaces of bounded distortion. We also prove several results on distorting the classical Tsirelson's space T and its relatives.