Collapsing theorem for Delaunay complexes in non-general position and symmetry

Abstract

Assuming general position, Bauer and Edelsbrunner show that the Cech complex collapses into the Delaunay complex for point sets in Euclidean space. By allowing non-unique solutions to certain minimal spheres, we bypass this assumption. Furthermore, using recursion, we prove an equivariant version of the collapsing theorem restricted to the plane assuming the point cloud has symmetry. We also discuss a simple symmetrical point cloud configuration in the plane that induces an indecomposable reducible representation on the barcode decomposition of a symmetric Delaunay complexes.