A step in the Delaunay mosaic of order k

Given a locally finite set $$ and an integer $$ , we consider the function $$ 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.