  We introduce a new diametral notion for points of the unit sphere of Banach
spaces, that naturally complements the notion of Delta-points, but is weaker
than the notion of Daugavet points. We prove that this notion can be used to
provide a new geometric characterization of the Daugavet property, as well as
to recover -- and even to provide new -- results about Daugavet points in
various contexts such as absolute sums of Banach spaces or projective tensor
products. Finally, we show that this notion leads to powerful new ideas for
renorming questions, and that those ideas can be combined with previous
constructions from the literature in order to renorm every infinite dimensional
Banach space with an unconditional Schauder basis to have a Daugavet point.
