We introduce relative versions of Daugavet-points and the Daugavet property,
where the Daugavet-behavior is localized inside of some supporting slice. These
points present striking similarities with Daugavet-points, but lie strictly
between the notions of Daugavet- and Δ-points. We provide a geometric
condition that a space with the Radon--Nikod\'{y}m property must satisfy in
order to be able to contain a relative Daugavet-point. We study relative
Daugavet-points in absolute sums of Banach spaces, and obtain positive
stability results under local polyhedrality of the underlying absolute norm. We
also get extreme differences between the relative Daugavet property, the
Daugavet property, and the diametral local diameter 2 property. Finally, we
study Daugavet- and Δ-points in subspaces of L1(μ)-spaces. We show
that the two notions coincide in the class of all Lipschitz-free spaces over
subsets of R-trees. We prove that the diametral local diameter 2
property and the Daugavet property coincide for arbitrary subspaces of
L1(μ), and that reflexive subspaces of L1(μ) do not contain
Δ-points. A subspace of L1[0,1] with a large subset of
Δ-points, but with no relative Daugavet-point, is constructed.Comment: 45 pages, 6 figure