Diametral notions for elements of the unit ball of a Banach space Martín Suárez, Miguel Perreau, Yoël Rueda Zoca, Abraham The first and third authors were supported by grant PID2021-122126NB-C31 funded by MICIU/AEI/10.13039/501100011033 and by ERDF/EU, by Junta de Andalucía I+D+i grants P20_00255 and FQM-185, and by “Maria de Maeztu” Excellence Unit IMAG (CEX2020-001105-M) funded by MICIU/AEI/10.13039/501100011033. The second named author was supported by the Estonian Research Council grant SJD58. We introduce extensions of Δ-points and Daugavet points in which slices are replaced by relatively weakly open subsets (super Δ-points and super Daugavet points) or by convex combinations of slices (ccs Δ-points and ccs Daugavet points). These notions represent the extreme opposite to denting points, points of continuity, and strongly regular points. We first give a general overview of these new concepts and provide some isometric consequences on the spaces. As examples: (1) If a Banach space contains a super Δ-point, then it does not admit an unconditional FDD (in particular, unconditional basis) with suppression constant smaller than 2. (2) If a real Banach space contains a ccs Δ-point, then it does not admit a one-unconditional basis. (3) If a Banach space contains a ccs Daugavet point, then every convex combination of slices of its unit ball has diameter 2. We next characterize the notions in some classes of Banach spaces, showing, for instance, that all the notions coincide in L1-predual spaces and that all the notions but ccs Daugavet points coincide in L1-spaces. We next comment on some examples which have previously appeared in the literature, and we provide some new intriguing examples: examples of super Δ-points which are as close as desired to strongly exposed points (hence failing to be Daugavet points in an extreme way); an example of a super Δ-point which is strongly regular (hence failing to be a ccs Δ-point in the strongest way); a super Daugavet point which fails to be a ccs Δ-point. The extensions of the diametral notions to points in the open unit ball and consequences on the spaces are also studied. Lastly, we investigate the Kuratowski measure of relatively weakly open subsets and of convex combinations of slices in the presence of super Δ-points or ccs Δ-points, as well as for spaces enjoying diameter-two properties. We conclude the paper with some open problems. 2024-06-03T12:00:07Z 2024-06-03T12:00:07Z 2024 journal article Published version: Martín Suárez, Miguel; Perreau, Yoël; Rueda Zoca, Abraham. Diametral notions for elements of the unit ball of a Banach space. Dissertationes Mathematicae 594 (2024), 1-61. DOI: 10.4064/dm230728-21-3 https://hdl.handle.net/10481/92266 10.4064/dm230728-21-3 eng http://creativecommons.org/licenses/by-nc-nd/3.0/ open access Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License Instytut Matematyczny PAN