Given a locally finite X⊆Rd and a radius r≥0, the k-fold cover of X and r consists of all points in Rd that have k or more points of X within distance r. We consider two filtrations—one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k—and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in Rd+1 whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module of Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.
Discrete and Computational Geometry
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 78818 Alpha), and by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through Grant No. I02979-N35 of the Austrian Science Fund (FWF) Open Access funding provided by the Institute of Science and Technology (IST Austria).
Edelsbrunner H, Osang GF. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 2021;65:1296–1313. doi:10.1007/s00454-021-00281-9
Edelsbrunner, H., & Osang, G. F. (2021). The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-021-00281-9
Edelsbrunner, Herbert, and Georg F Osang. “The Multi-Cover Persistence of Euclidean Balls.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-021-00281-9.
H. Edelsbrunner and G. F. Osang, “The multi-cover persistence of Euclidean balls,” Discrete and Computational Geometry, vol. 65. Springer Nature, pp. 1296–1313, 2021.
Edelsbrunner H, Osang GF. 2021. The multi-cover persistence of Euclidean balls. Discrete and Computational Geometry. 65, 1296–1313.
Edelsbrunner, Herbert, and Georg F. Osang. “The Multi-Cover Persistence of Euclidean Balls.” Discrete and Computational Geometry, vol. 65, Springer Nature, 2021, pp. 1296–1313, doi:10.1007/s00454-021-00281-9.
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