About smooth and non-poor subspaces of Daugavet spaces

TL;DR

This paper explores the properties of Daugavet spaces, highlighting examples with Gâteaux differentiability, and investigates subspaces of C[0,1] related to the Daugavet property and slice diameter properties.

Contribution

It provides new examples of Daugavet spaces with differentiability properties and analyzes subspace structures affecting the Daugavet and slice diameter properties.

Findings

A non-complete Daugavet space can have a Gâteaux differentiable norm at all nonzero points.

Dual norms of Daugavet spaces are not Gâteaux differentiable at any point.

Certain subspaces of C[0,1], like quasilacunary M"untz spaces, do not have the Daugavet property, but preserve the slice diameter two property.

Abstract

We discuss an example of a non-complete normed space with the Daugavet property such that the norm is G\^ateaux differentiable at every nonzero point. In contrast, we note that the dual norm of a normed space with the Daugavet property is not G\^ateaux differentiable at any point. Furthermore, we show that quasilacunary M\"untz spaces form a natural class of subspaces of $C[0,1]$, isomorphic to $c_0$, for which the corresponding quotient spaces fail to have the Daugavet property. At the same time, the slice diameter two property is preserved under this construction.