The Daugavet property for Sobolev spaces over the plane

TL;DR

This paper investigates the Daugavet property in Sobolev spaces over the plane, showing it holds for a specific norm but not for the usual one, revealing nuanced geometric features of these function spaces.

Contribution

It demonstrates the Daugavet property for $W^{1,1}(R^2)$ with a gradient-based norm and identifies the absence of the slice diameter two property with the standard Sobolev norm.

Findings

Daugavet property holds for $W^{1,1}(R^2)$ with the gradient norm.

Fails to have the slice diameter two property with the usual Sobolev norm.

Highlights geometric differences based on norm choice in Sobolev spaces.

Abstract

We show that $W^{1,1}(\mathbb{R}^2)$ has the Daugavet property when endowed with the norm induced by the $L^1$-norm of the gradient, but fails to have the slice diameter two property when equipped with the usual Sobolev norm.