The Morse theory of \v{C}ech and Delaunay filtrations
Ulrich Bauer, Herbert Edelsbrunner · arXiv (Cornell University) · 2013
Given a finite set of points in $\mathbb R^n$ and a positive radius, we consider the \v{C}ech, Delaunay-\v{C}ech, alpha, and wrap complexes as examples of a generalized discrete Morse theory. We prove that the latter three are simple-homotopy equivalent, and the same is true for their weighted versions. Our results have applications in high-dimensional data analysis and in the reconstruction of shapes from sampled data.