On the geometry of higher order Schreier spaces

Abstract: For each countable ordinal $\alpha$ let $\mathcal{S}{\alpha}$ be the Schreier set of order $\alpha$ and $X{\mathcal{S}\alpha}$ be the corresponding Schreier space of order $\alpha$. In this paper we prove several new properties of these spaces. 1) If $\alpha$ is non-zero then $X{\mathcal{S}\alpha}$ possesses the $\lambda$-property of R. Aron and R. Lohman and is a $(V)$-polyhedral spaces in the sense on V. Fonf and L. Vesely. 2) If $\alpha$ is non-zero and $1<p<\infty$ then the $p$-convexification $X^{p}{\mathcal{S}\alpha}$ possesses the uniform $\lambda$-property of R. Aron and R. Lohman. 3) For each countable ordinal $\alpha$ the space $X^*{\mathcal{S}\alpha}$ has the $\lambda$-property. 4) For $n\in \mathbb{N}$, if $U:X{\mathcal{S}n}\to X{\mathcal{S}n}$ is an onto linear isometry then $Ue_i = \pm e_i$ for each $i \in \mathbb{N}$. Consequently, these spaces are light in the sense of Megrelishvili. The fact that for non-zero $\alpha$, $X{\mathcal{S}\alpha}$ is $(V)$-polyhedral and has the $\lambda$-property implies that each $X{\mathcal{S}_\alpha}$ is an example of space solving a problem of J. Lindenstrauss from 1966. The first example of such a space was given by C. De Bernardi in 2017 using a renorming of $c_0$.