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$\ell _{1}$ spreading models and the Lebesgue property of Banach spaces
We construct a reflexive Banach space 
$X_{\mathcal {D}}$ with an unconditional basis such that all spreading models admitted by normalized block sequences in 
$X_{\mathcal {D}}$ are uniformly equivalent to the unit vector basis of 
$\ell _1$, yet every infinite-dimensional closed subspace of 
$X_{\mathcal {D}}$ fails the Lebesgue property. This is a new result in a program initiated by Odell in 2002 concerning the strong separation of asymptotic properties in Banach spaces.
We show that for a bounded subset 
$A$ of the 
$L_{1}(\unicode[STIX]{x1D707})$ space with finite measure 
$\unicode[STIX]{x1D707}$, the measure of weak noncompactness of 
$A$ based on the convex separation of sequences coincides with the measure of deviation from the Banach–Saks property expressed by the arithmetic separation of sequences. A similar result holds for a related quantity with the alternating signs Banach–Saks property. The results provide a geometric and quantitative extension of Szlenk’s theorem saying that every weakly convergent sequence in the Lebesgue space 
$L_{1}$ has a subsequence whose arithmetic means are norm convergent.
