Geodesic
The complete separation of the two finer asymptotic $\boldsymbol {\ell _{p}}$ structures for $ \boldsymbol {1\le p}$$ \boldsymbol {\infty }$
Forum of Mathematics, Sigma, Tome 10 (2022)

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For $1\le p \infty $, we present a reflexive Banach space $\mathfrak {X}^{(p)}_{\text {awi}}$, with an unconditional basis, that admits $\ell _p$ as a unique asymptotic model and does not contain any Asymptotic $\ell _p$ subspaces. Freeman et al., Trans. AMS. 370 (2018), 6933–6953 have shown that whenever a Banach space not containing $\ell _1$, in particular a reflexive Banach space, admits $c_0$ as a unique asymptotic model, then it is Asymptotic $c_0$. These results provide a complete answer to a problem posed by Halbeisen and Odell [Isr. J. Math. 139 (2004), 253–291] and also complete a line of inquiry of the relation between specific asymptotic structures in Banach spaces, initiated in a previous paper by the first and fourth authors. For the definition of $\mathfrak {X}^{(p)}_{\text {awi}}$, we use saturation with asymptotically weakly incomparable constraints, a new method for defining a norm that remains small on a well-founded tree of vectors which penetrates any infinite dimensional closed subspace. 
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     author = {Spiros A. Argyros and Alexandros Georgiou and Antonis Manoussakis and Pavlos Motakis},
     title = {The complete separation of the two finer asymptotic $\boldsymbol {\ell _{p}}$ structures for $ \boldsymbol {1\le p}$<$ \boldsymbol {\infty }$},
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Spiros A. Argyros; Alexandros Georgiou; Antonis Manoussakis; Pavlos Motakis. The complete separation of the two finer asymptotic $\boldsymbol {\ell _{p}}$ structures for $ \boldsymbol {1\le p}$<$ \boldsymbol {\infty }$. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.101

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