For each countable ordinal α, let be the Schreier set of order α and be the corresponding Schreier space of order α. In this paper, we prove several new properties of these spaces.
1. If α is nonzero, then possesses the λ-property of Aron and Lohman and is a -polyhedral space in the sense of Fonf and Vesely.
2. If α is nonzero and , then the p-convexification possesses the uniform λ-property of Aron and Lohman.
3. For each countable ordinal α, the space has the λ-property.
4. For , if is an onto linear isometry, then for each . Consequently, these spaces are light in the sense of Megrelishvili.
The fact that for nonzero α, is polyhedral and has the λ-property implies that each is an example of a space that solves a problem of Lindenstrauss from 1966 about the existence of a polyhedral infinite dimensional Banach space whose unit ball is the closed convex hull of its extreme points. The first example of such a space was given by De Bernardi in 2017 using a renorming of .
"On the geometry of higher order Schreier spaces." Illinois J. Math. 65 (1) 47 - 69, April 2021. https://doi.org/10.1215/00192082-8827623