We show that W1,1(R2) has the Daugavet property when endowed with the norm induced by the L1-norm of the gradient, but fails to have the slice diameter two property when equipped with the usual Sobolev norm.
@article{arxiv.2601.21486,
title = {The Daugavet property for Sobolev spaces over the plane},
author = {Samir Hamad},
journal= {arXiv preprint arXiv:2601.21486},
year = {2026}
}