The Morse theory of vCech and Delaunay complexes

Given a finite set of points in mathbb Rn and a radius parameter, we study the vCech, Delaunay-vCech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the vCech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.