  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 $c_0$.
