A relative version of Daugavet-points and the Daugavet property

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 $\Delta$-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 $\Delta$-points in subspaces of $L_1(\mu)$-spaces. We show that the two notions coincide in the class of all Lipschitz-free spaces over subsets of $\mathbb{R}$-trees. We prove that the diametral local diameter 2 property and the Daugavet property coincide for arbitrary subspaces of $L_1(\mu)$, and that reflexive subspaces of $L_1(\mu)$ do not contain $\Delta$-points. A subspace of $L_1[0,1]$ with a large subset of $\Delta$-points, but with no relative Daugavet-point, is constructed.