Asymptotic normality for random polytopes in non-Euclidean geometries
HTML articles powered by AMS MathViewer
- by Florian Besau and Christoph Thäle PDF
- Trans. Amer. Math. Soc. 373 (2020), 8911-8941 Request permission
Abstract:
Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and central limit theorems in Lutwak’s dual Brunn–Minkowski theory are established. The results follow from a central limit theorem for weighted random polytopes in Euclidean spaces. In the background are Stein’s method for normal approximation and geometric properties of weighted floating bodies.References
- F. Affentranger, The convex hull of random points with spherically symmetric distributions, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), no. 3, 359–383 (1993). MR 1231058
- David Alonso-Gutiérrez, Martin Henk, and María A. Hernández Cifre, A characterization of dual quermassintegrals and the roots of dual Steiner polynomials, Adv. Math. 331 (2018), 565–588. MR 3804685, DOI 10.1016/j.aim.2018.04.008
- J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 1–48. MR 2132656, DOI 10.4171/prims/123
- Imre Bárány, Random polytopes in smooth convex bodies, Mathematika 39 (1992), no. 1, 81–92. MR 1176473, DOI 10.1112/S0025579300006872
- Imre Bárány, Random polytopes, convex bodies, and approximation, Stochastic geometry, Lecture Notes in Math., vol. 1892, Springer, Berlin, 2007, pp. 77–118. MR 2327291, DOI 10.1007/978-3-540-38175-4_{2}
- I. Bárány, F. Fodor, and V. Vígh, Intrinsic volumes of inscribed random polytopes in smooth convex bodies, Adv. in Appl. Probab. 42 (2010), no. 3, 605–619. MR 2779551, DOI 10.1239/aap/1282924055
- Imre Bárány, Daniel Hug, Matthias Reitzner, and Rolf Schneider, Random points in halfspheres, Random Structures Algorithms 50 (2017), no. 1, 3–22. MR 3583024, DOI 10.1002/rsa.20644
- Imre Bárány and Matthias Reitzner, On the variance of random polytopes, Adv. Math. 225 (2010), no. 4, 1986–2001. MR 2680197, DOI 10.1016/j.aim.2010.04.012
- Imre Bárány and Christoph Thäle, Intrinsic volumes and Gaussian polytopes: the missing piece of the jigsaw, Doc. Math. 22 (2017), 1323–1335. MR 3722561
- Andreas Bernig, The isoperimetrix in the dual Brunn-Minkowski theory, Adv. Math. 254 (2014), 1–14. MR 3161087, DOI 10.1016/j.aim.2013.12.020
- F. Besau, T. Hack, P. Pivovarov, and F. E. Schuster, Spherical centroid bodies, preprint (2019), arXiv:1902.10614.
- F. Besau, S. Hoehner, and G. Kur, Intrinsic and dual volume deviations of convex bodies and polytopes, Int. Math. Res. Not. IMRN (2019), 58 pp. https://doi.org/10.1093/imrn/rnz277
- Florian Besau, Monika Ludwig, and Elisabeth M. Werner, Weighted floating bodies and polytopal approximation, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7129–7148. MR 3841844, DOI 10.1090/tran/7233
- Florian Besau and Elisabeth M. Werner, The spherical convex floating body, Adv. Math. 301 (2016), 867–901. MR 3539392, DOI 10.1016/j.aim.2016.07.001
- Florian Besau and Elisabeth M. Werner, The floating body in real space forms, J. Differential Geom. 110 (2018), no. 2, 187–220. MR 3861810, DOI 10.4310/jdg/1538791243
- V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
- Károly J. Böröczky Jr., Lars Michael Hoffmann, and Daniel Hug, Expectation of intrinsic volumes of random polytopes, Period. Math. Hungar. 57 (2008), no. 2, 143–164. MR 2469601, DOI 10.1007/s10998-008-8143-4
- Károly J. Böröczky, Ferenc Fodor, and Daniel Hug, The mean width of random polytopes circumscribed around a convex body, J. Lond. Math. Soc. (2) 81 (2010), no. 2, 499–523. MR 2603007, DOI 10.1112/jlms/jdp077
- K. J. Böröczky, F. Fodor, M. Reitzner, and V. Vígh, Mean width of random polytopes in a reasonably smooth convex body, J. Multivariate Anal. 100 (2009), no. 10, 2287–2295. MR 2560369, DOI 10.1016/j.jmva.2009.07.003
- Károly J. Böröczky, Martin Henk, and Hannes Pollehn, Subspace concentration of dual curvature measures of symmetric convex bodies, J. Differential Geom. 109 (2018), no. 3, 411–429. MR 3825606, DOI 10.4310/jdg/1531188189
- J. S. Brauchart, A. B. Reznikov, E. B. Saff, I. H. Sloan, Y. G. Wang, and R. S. Womersley, Random point sets on the sphere—hole radii, covering, and separation, Exp. Math. 27 (2018), no. 1, 62–81. MR 3750928, DOI 10.1080/10586458.2016.1226209
- Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of isotropic convex bodies, Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014. MR 3185453, DOI 10.1090/surv/196
- P. Calka, A. Chapron, and N. Enriquez, Mean asymptotics for a Poisson–Voronoi cell on a Riemannian manifold, preprint (2018), arXiv:1807.09043.
- Sourav Chatterjee, A new method of normal approximation, Ann. Probab. 36 (2008), no. 4, 1584–1610. MR 2435859, DOI 10.1214/07-AOP370
- Christian Deuss, Julia Hörrmann, and Christoph Thäle, A random cell splitting scheme on the sphere, Stochastic Process. Appl. 127 (2017), no. 5, 1544–1564. MR 3630235, DOI 10.1016/j.spa.2016.08.010
- R. J. Gardner, The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities, Adv. Math. 216 (2007), no. 1, 358–386. MR 2353261, DOI 10.1016/j.aim.2007.05.018
- R. J. Gardner, Eva B. Vedel Jensen, and A. Volčič, Geometric tomography and local stereology, Adv. in Appl. Math. 30 (2003), no. 3, 397–423. MR 1973951, DOI 10.1016/S0196-8858(02)00502-X
- Christoph Haberl and Lukas Parapatits, The centro-affine Hadwiger theorem, J. Amer. Math. Soc. 27 (2014), no. 3, 685–705. MR 3194492, DOI 10.1090/S0894-0347-2014-00781-5
- Yong Huang, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), no. 2, 325–388. MR 3573332, DOI 10.1007/s11511-016-0140-6
- Julia Hörrmann, Daniel Hug, Matthias Reitzner, and Christoph Thäle, Poisson polyhedra in high dimensions, Adv. Math. 281 (2015), 1–39. MR 3366836, DOI 10.1016/j.aim.2015.03.025
- Daniel Hug, Contributions to affine surface area, Manuscripta Math. 91 (1996), no. 3, 283–301. MR 1416712, DOI 10.1007/BF02567955
- D. Hug and A. Reichenbacher, Geometric inequalities, stability results and Kendall’s problem in spherical space, preprint (2017), arXiv:1709.06522.
- Daniel Hug and Christoph Thäle, Splitting tessellations in spherical spaces, Electron. J. Probab. 24 (2019), Paper No. 24, 60. MR 3933203, DOI 10.1214/19-EJP267
- Zakhar Kabluchko, Alexander Marynych, Daniel Temesvari, and Christoph Thäle, Cones generated by random points on half-spheres and convex hulls of Poisson point processes, Probab. Theory Related Fields 175 (2019), no. 3-4, 1021–1061. MR 4026612, DOI 10.1007/s00440-019-00907-3
- Raphaël Lachièze-Rey and Giovanni Peccati, New Berry-Esseen bounds for functionals of binomial point processes, Ann. Appl. Probab. 27 (2017), no. 4, 1992–2031. MR 3693518, DOI 10.1214/16-AAP1218
- Kurt Leichtweiß, Zur Affinoberfläche konvexer Körper, Manuscripta Math. 56 (1986), no. 4, 429–464 (German). MR 860732, DOI 10.1007/BF01168504
- Monika Ludwig and Matthias Reitzner, A characterization of affine surface area, Adv. Math. 147 (1999), no. 1, 138–172. MR 1725817, DOI 10.1006/aima.1999.1832
- Monika Ludwig and Matthias Reitzner, A classification of $\textrm {SL}(n)$ invariant valuations, Ann. of Math. (2) 172 (2010), no. 2, 1219–1267. MR 2680490, DOI 10.4007/annals.2010.172.1223
- Erwin Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531–538. MR 380631
- Erwin Lutwak, Mean dual and harmonic cross-sectional measures, Ann. Mat. Pura Appl. (4) 119 (1979), 139–148. MR 551220, DOI 10.1007/BF02413172
- Erwin Lutwak, Extended affine surface area, Adv. Math. 85 (1991), no. 1, 39–68. MR 1087796, DOI 10.1016/0001-8708(91)90049-D
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ dual curvature measures, Adv. Math. 329 (2018), 85–132. MR 3783409, DOI 10.1016/j.aim.2018.02.011
- H. Maehara and H. Martini, An analogue of Sylvester’s four-point problem on the sphere, Acta Math. Hungar. 155 (2018), no. 2, 479–488. MR 3831312, DOI 10.1007/s10474-018-0814-y
- Marc Troyanov, On the origin of Hilbert geometry, Handbook of Hilbert geometry, IRMA Lect. Math. Theor. Phys., vol. 22, Eur. Math. Soc., Zürich, 2014, pp. 383–389. MR 3329888
- Matthias Reitzner, Poisson point processes: large deviation inequalities for the convex distance, Electron. Commun. Probab. 18 (2013), no. 96, 7. MR 3151752, DOI 10.1214/ECP.v18-2851
- Mathew D. Penrose and J. E. Yukich, Limit theory for point processes in manifolds, Ann. Appl. Probab. 23 (2013), no. 6, 2161–2211. MR 3127932, DOI 10.1214/12-AAP897
- C. M. Petty, Affine isoperimetric problems, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 113–127. MR 809198, DOI 10.1111/j.1749-6632.1985.tb14545.x
- Matthias Reitzner, Stochastic approximation of smooth convex bodies, Mathematika 51 (2004), no. 1-2, 11–29 (2005). MR 2220208, DOI 10.1112/S0025579300015473
- Matthias Reitzner, Central limit theorems for random polytopes, Probab. Theory Related Fields 133 (2005), no. 4, 483–507. MR 2197111, DOI 10.1007/s00440-005-0441-8
- Ross M. Richardson, Van H. Vu, and Lei Wu, An inscribing model for random polytopes, Discrete Comput. Geom. 39 (2008), no. 1-3, 469–499. MR 2383772, DOI 10.1007/s00454-007-9012-3
- Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326, DOI 10.1007/978-3-540-78859-1
- Carsten Schütt and Elisabeth Werner, The convex floating body, Math. Scand. 66 (1990), no. 2, 275–290. MR 1075144, DOI 10.7146/math.scand.a-12311
- Carsten Schütt and Elisabeth Werner, Homothetic floating bodies, Geom. Dedicata 49 (1994), no. 3, 335–348. MR 1270561, DOI 10.1007/BF01264033
- Carsten Schütt and Elisabeth Werner, Polytopes with vertices chosen randomly from the boundary of a convex body, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 241–422. MR 2083401, DOI 10.1007/978-3-540-36428-3_{1}9
- Christoph Thäle, Central limit theorem for the volume of random polytopes with vertices on the boundary, Discrete Comput. Geom. 59 (2018), no. 4, 990–1000. MR 3802312, DOI 10.1007/s00454-017-9862-2
- Christoph Thäle, Nicola Turchi, and Florian Wespi, Random polytopes: central limit theorems for intrinsic volumes, Proc. Amer. Math. Soc. 146 (2018), no. 7, 3063–3071. MR 3787367, DOI 10.1090/proc/14000
- A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
- Marc Troyanov, Funk and Hilbert geometries from the Finslerian viewpoint, Handbook of Hilbert geometry, IRMA Lect. Math. Theor. Phys., vol. 22, Eur. Math. Soc., Zürich, 2014, pp. 69–110. MR 3329877
- N. Turchi and F. Wespi, Limit theorems for random polytopes with vertices on convex surfaces, Adv. in Appl. Probab. 50 (2018), no. 4, 1227–1245. MR 3881117, DOI 10.1017/apr.2018.58
- V. H. Vu, Sharp concentration of random polytopes, Geom. Funct. Anal. 15 (2005), no. 6, 1284–1318. MR 2221249, DOI 10.1007/s00039-005-0541-8
- Elisabeth Werner, The $p$-affine surface area and geometric interpretations, Rend. Circ. Mat. Palermo (2) Suppl. 70 (2002), 367–382. IV International Conference in “Stochastic Geometry, Convex Bodies, Empirical Measures $\&$ Applications to Engineering Science”, Vol. II (Tropea, 2001). MR 1962608
Additional Information
- Florian Besau
- Affiliation: Department of Mathematics, Vienna University of Technology, Vienna, Austria
- MR Author ID: 1174501
- ORCID: 0000-0002-6596-6127
- Email: florian.besau@tuwien.ac.at
- Christoph Thäle
- Affiliation: Department of Mathematics, Ruhr University Bochum, Bochum, Germany
- Email: christoph.thaele@rub.de
- Received by editor(s): September 12, 2019
- Received by editor(s) in revised form: May 24, 2020
- Published electronically: October 5, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8911-8941
- MSC (2010): Primary 52A22, 52A55; Secondary 60D05, 60F05
- DOI: https://doi.org/10.1090/tran/8217
- MathSciNet review: 4177280