On the geometry of higher order Schreier spaces

For each countable ordinal α, let Sα be the Schreier set of order α and XSα corresponding space α. In this paper, we prove several new properties these spaces. 1. If is nonzero, then XSαpossesses λ-property Aron Lohman a (V)-polyhedral in sense Fonf Vesely. 2. nonzero 1<p<∞, p-convexification p possesses uniform Lohman. 3. ∗ has λ-property. 4. n∈N, if T:XSn→XSnis an onto linear isometry, Tei=±ei for i∈N. Consequently, spaces are light Megrelishvili. The fact that polyhedral implies example solves problem Lindenstrauss from 1966 about existence infinite dimensional Banach whose unit ball closed convex hull its extreme points. first such was given by De Bernardi 2017 using renorming c0.