A step in the Delaunay mosaic of order .

Given a locally finite set and an integer , we consider the function on the dual of the order- Voronoi tessellation, whose sublevel sets generalize the notion of alpha shapes from order-1 to order- (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 -fold covers with balls in Euclidean space.