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Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  08 September 2017

Herbert Edelsbrunner ,
Anton Nikitenko  and
Matthias Reitzner
Herbert Edelsbrunner*
Affiliation:
IST Austria
Anton Nikitenko*
Affiliation:
IST Austria
Matthias Reitzner*
Affiliation:
University of Osnabrück
*
* Postal address: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
* Postal address: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
**** Postal address: Mathematics Department, University of Osnabrück, Albrechtstrasse 28a, 49076 Osnabrück, Germany. Email address: [email protected]
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Abstract

Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4.

Keywords

Poisson point processDelaunay mosaicdiscrete Morse theorycritical simplicesintervalsstochastic geometryintegral geometrytypical simplex

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 68U05: Computer graphics; computational geometry
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 745 - 767
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Bauer, U. and Edelsbrunner, H. (2017). The Morse theory of Čech and Delaunay complexes. Trans. Amer. Math. Soc. 369, 3741–3762. Google Scholar
[2] Baumstark, V. and Last, G. (2009). Gamma distributions for stationary Poisson flat processes. Adv. Appl. Prob. 41, 911–939. Google Scholar
[3] Bobrowski, O. and Adler, R. J. (2014). Distance functions, critical points, and the topology of random Čech complexes. Homology Homotopy Appl. 16, 311–344. CrossRefGoogle Scholar
[4] Bobrowski, O. and Weinberger, S. (2017). On the vanishing of homology in random Čech complexes. Random Structures Algorithms 51, 14–51. Google Scholar
[5] Bobrowski, O., Kahle, M. and Skraba, P. (2016). Maximally persistent cycles in random geometric complexes. To appear in Ann. Appl. Prob. Google Scholar
[6] Carlsson, G. (2009). Topology and data. Bull. Amer. Math. Soc. (N.S.) 46, 255–308. Google Scholar
[7] Chenavier, N. (2014). A general study of extremes of stationary tessellations with applications. Stoch. Process. Appl. 124, 2917–2953. CrossRefGoogle Scholar
[8] Decreusefond, L., Ferraz, E., Randriambololona, H. and Vergne, A. (2014). Simplicial homology of random configurations. Adv. Appl. Prob. 46, 325–347. CrossRefGoogle Scholar
[9] Dey, T. K. (2007). Curve and Surface Reconstruction. Algorithms with Mathematical Analysis. Cambridge University Press. Google Scholar
[10] Drton, M., Massam, H. and Olkin, I. (2008). Moments of minors of Wishart matrices. Ann. Statist. 36, 2261–2283. Google Scholar
[11] Edelsbrunner, H. (2001). Geometry and Topology for Mesh Generation. Cambridge University Press. CrossRefGoogle Scholar
[12] Edelsbrunner, H. (2003). Surface reconstruction by wrapping finite sets in space. In Discrete and Computational Geometry, Springer, Berlin, pp. 379–404. Google Scholar
[13] Edelsbrunner, H. and Harer, J. L. (2010). Computational Topology: An Introduction. American Mathematical Society, Providence, RI. Google Scholar
[14] Edelsbrunner, H. and Mücke, E. P. (1994). Three-dimensional alpha shapes. ACM Trans. Graphics 13, 43–72. Google Scholar
[15] Forman, R. (1998). Morse theory for cell complexes. Adv. Math. 134, 90–145. Google Scholar
[16] Freij, R. (2009). Equivariant discrete Morse theory. Discrete Math. 309, 3821–3829. Google Scholar
[17] Kahle, M. (2011). Random geometric complexes. Discrete Comput. Geom. 45, 553–573. Google Scholar
[18] Kahle, M. (2014). Topology of random simplicial complexes: a survey. In Algebraic Topology: Applications and New Directions (Contemp. Math. 620), American Mathematical Society, Providence, RI, pp. 201–222. Google Scholar
[19] Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press. Google Scholar
[20] Miles, R. E. (1969). Poisson flats in Euclidean spaces. I. A finite number of random uniform flats. Adv. Appl. Prob. 1, 211–237. Google Scholar
[21] Miles, R. E. (1970). On the homogeneous planar Poisson point process. Math. Biosci. 6, 85–127. Google Scholar
[22] Miles, R. E. (1971). Isotropic random simplices. Adv. Appl. Prob. 3, 353–382. CrossRefGoogle Scholar
[23] Miles, R. E. (1974). A synopsis of 'Poisson flats in Euclidean spaces'. In Stochastic Geometry, eds E. F. Harding and D. G. Kendall, John Wiley, New York, pp. 202–227. Google Scholar
[24] Møller, J. (1989). Random tessellations in ℝ d . Adv. Appl. Prob 21, 37–73. CrossRefGoogle Scholar
[25] Olver, F. W. J. (1997). Asymptotics and Special Functions. A. K. Peters, Wellesley, MA. Google Scholar
[26] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin. CrossRefGoogle Scholar
[27] Wendel, J. G. (1962). A problem in geometric probability. Math. Scand. 11, 109–111. CrossRefGoogle Scholar