--- res: bibo_abstract: - "Given a locally finite set \U0001D44B⊆ℝ\U0001D451 and an integer \U0001D458≥0, we consider the function \U0001D430\U0001D458:Del\U0001D458(\U0001D44B)→ℝ on the dual of the order-k Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order-k (Edelsbrunner et al. in IEEE Trans Inf Theory IT-29:551–559, 1983; Krasnoshchekov and Polishchuk in Inf Process Lett 114:76–83, 2014). While this function is not necessarily generalized discrete Morse, in the sense of Forman (Adv Math 134:90–145, 1998) and Freij (Discrete Math 309:3821–3829, 2009), we prove that it satisfies similar properties so that its increments can be meaningfully classified into critical and non-critical steps. This result extends to the case of weighted points and sheds light on k-fold covers with balls in Euclidean space.@eng" bibo_authorlist: - foaf_Person: foaf_givenName: Herbert foaf_name: Edelsbrunner, Herbert foaf_surname: Edelsbrunner foaf_workInfoHomepage: http://www.librecat.org/personId=3FB178DA-F248-11E8-B48F-1D18A9856A87 orcid: 0000-0002-9823-6833 - foaf_Person: foaf_givenName: Anton foaf_name: Nikitenko, Anton foaf_surname: Nikitenko foaf_workInfoHomepage: http://www.librecat.org/personId=3E4FF1BA-F248-11E8-B48F-1D18A9856A87 - foaf_Person: foaf_givenName: Georg F foaf_name: Osang, Georg F foaf_surname: Osang foaf_workInfoHomepage: http://www.librecat.org/personId=464B40D6-F248-11E8-B48F-1D18A9856A87 bibo_doi: 10.1007/s00022-021-00577-4 bibo_issue: '1' bibo_volume: 112 dct_date: 2021^xs_gYear dct_isPartOf: - http://id.crossref.org/issn/00472468 - http://id.crossref.org/issn/14208997 dct_language: eng dct_publisher: Springer Nature@ dct_title: A step in the Delaunay mosaic of order k@ ...