![]() Discrete Morse theory was introduced by Forman and generalized to allow for collapses across more than one dimension by Freij. Three-dimensional alpha shapes have found ample applications in shape modeling and in the analysis of biomolecules. The radius function on the Delaunay mosaic was first introduced in, along with its sublevel sets, which are the alpha shapes of the given points. The Voronoi tessellation and the dual Delaunay mosaic are classic topics in discrete geometry and go back at least to the seminal papers by Voronoi and by Delaunay. We prove new results on these tessellations and mosaics by exploiting the structural properties of these functions. We weave the two strands of inquiry together by studying the continuous and discrete radius functions that define Voronoi tessellations and Delaunay mosaics for weighted points not necessarily in general position. How do we relax the theory to allow for non-generic data? Related to this question is the symmetry between Voronoi tessellations and Delaunay mosaics introduced in this paper, which appears when we have weights, and non-generic data is essential to realize this symmetry. ![]() In contrast, without the general position assumption, the mosaics are not simplicial and the radius functions are not generalized discrete Morse. For points in general position, the radius function is generalized discrete Morse. The starting point for the work reported in this paper is the role of the general position assumption in the construction of Delaunay mosaics, and more specifically of their radius functions. ![]()
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