On the geometry of higher order Schreier spaces

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 $1<p<\mathrm{\infty }$ then the $p$-convexification $X_{{\mathcal{S}}_{\alpha }}^p$ possesses the uniform $\lambda$-property of R. Aron and R. Lohman. 3) For each countable ordinal $\alpha$ the space $X_{{\mathcal{S}}_{\alpha }}^{\ast }$ 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 $U{e}_i=\pm e_i$ 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 $c_0$.