  We introduce extensions of $\Delta$-points and Daugavet points in which
slices are replaced by relative weakly open subsets (super $\Delta$-points and
super Daugavet points) or by convex combinations of slices (ccs $\Delta$-points
and ccs Daugavet points). We first give a general overview on these new
concepts and provide some isometric consequences on the spaces. As examples: if
a Banach space contains a super $\Delta$-point, then it does not admit an
unconditional FDD with suppression constant smaller than two; if a real Banach
space contains a ccs $\Delta$-point, then it does not admit a one-unconditional
basis; if a Banach space contains a ccs Daugavet point, then every convex
combination of slices of its unit ball has diameter two. We next characterize
the notions in some classes of Banach spaces showing, for instance, that all
the notions coincide in $L_1$-predual spaces and that all the notions but ccs
Daugavet points coincide in $L_1$-spaces. We next remark on some examples which
have previously appeared in the literature and provide some new intriguing
examples: examples of super $\Delta$-points which are as closed as desired to
strongly exposed points (hence failing to be Daugavet points in an extreme
way); an example of a super $\Delta$-point which is strongly regular (hence
failing to be a ccs $\Delta$-point in the strongest way); a super Daugavet
point which fails to be a ccs $\Delta$-point. The extensions of the diametral
notions to point in the open unit ball and the consequences on the spaces are
also studied. Last, we investigate the Kuratowski measure of relative weakly
open subsets and of convex combinations of slices in the presence of super
$\Delta$-points or ccs $\Delta$-points, as well as for spaces enjoying diameter
2 properties. We conclude the paper with a section on open problems.
