Zakhar Kabluchko Dissertation Sample

   

 

 

 

Monotonicity of expected -vectors for projections of regular polytopes


Authors:Zakhar Kabluchko and Christoph Thäle
Journal: Proc. Amer. Math. Soc. 146 (2018), 1295-1303
MSC (2010): Primary 52A22, 60D05; Secondary 52B11, 52A20, 51M20
DOI: https://doi.org/10.1090/proc/13827
Published electronically: October 6, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an -dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project onto a random, uniformly distributed linear subspace of dimension . We prove that the expected number of -dimensional faces of the resulting random polytope is an increasing function of . As a corollary, we show that the expected number of -faces of the Gaussian polytope is an increasing function of the number of points used to generate the polytope. Similar results are obtained for the symmetric Gaussian polytope and the Gaussian zonotope.


  • [1] Fernando Affentranger and Rolf Schneider, Random projections of regular simplices, Discrete Comput. Geom. 7 (1992), no. 3, 219-226. MR 1149653, https://doi.org/10.1007/BF02187839
  • [2] Yuliy M. Baryshnikov and Richard A. Vitale, Regular simplices and Gaussian samples, Discrete Comput. Geom. 11 (1994), no. 2, 141-147. MR 1254086, https://doi.org/10.1007/BF02574000
  • [3] Mareen Beermann, Random Polytopes, Ph.D. thesis, University of Osnabrück, 2015, Available at: .
  • [4] Mareen Beermann and Matthias Reitzner, Monotonicity of functionals of random polytopes, In: G. Ambrus, K. Böröczky, and Z. Füredi, (eds.): Discrete Geometry and Convexity, pp. 23-28, A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, 2017, preprint available at http://arxiv.org/abs/1706.08342.
  • [5] Marcel Berger, Geometry. II, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882916
  • [6] Ulrich Betke and Martin Henk, Intrinsic volumes and lattice points of crosspolytopes, Monatsh. Math. 115 (1993), no. 1-2, 27-33. MR 1223242, https://doi.org/10.1007/BF01311208
  • [7] Johannes Böhm and Eike Hertel, Polyedergeometrie in -dimensionalen Räumen konstanter Krümmung, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series], vol. 70, Birkhäuser Verlag, Basel-Boston, Mass., 1981 (German). MR 626823
  • [8] Gilles Bonnet, Julian Grote, Daniel Temesvari, Christoph Thäle, Nicola Turchi, and Florian Wespi, Monotonicity of facet numbers of random convex hulls, J. Math. Anal. Appl. 455 (2017), no. 2, 1351-1364. MR 3671230, https://doi.org/10.1016/j.jmaa.2017.06.054
  • [9] Károly Böröczky Jr. and Martin Henk, Random projections of regular polytopes, Arch. Math. (Basel) 73 (1999), no. 6, 465-473. MR 1725183, https://doi.org/10.1007/s000130050424
  • [10] Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, and Matthias Reitzner, The monotonicity of -vectors of random polytopes, Electron. Commun. Probab. 18 (2013), no. 23, 8. MR 3044471, https://doi.org/10.1214/ECP.v18-2469
  • [11] David L. Donoho and Jared Tanner, Counting faces of randomly projected polytopes when the projection radically lowers dimension, J. Amer. Math. Soc. 22 (2009), no. 1, 1-53. MR 2449053, https://doi.org/10.1090/S0894-0347-08-00600-0
  • [12] David L. Donoho and Jared Tanner, Counting the faces of randomly-projected hypercubes and orthants, with applications, Discrete Comput. Geom. 43 (2010), no. 3, 522-541. MR 2587835, https://doi.org/10.1007/s00454-009-9221-z
  • [13] Branko Grünbaum, Convex polytopes, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR 1976856
  • [14] Daniel Hug, Götz Olaf Munsonius, and Matthias Reitzner, Asymptotic mean values of Gaussian polytopes, Beiträge Algebra Geom. 45 (2004), no. 2, 531-548. MR 2093024
  • [15] K. Leichtweiss, Konvexe Mengen, Hochschulbücher für Mathematik [University Books for Mathematics], 81, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980 (German). MR 559138
  • [16] Harold Ruben, On the geometrical moments of skew-regular simplices in hyperspherical space, with some applications in geometry and mathematical statistics, Acta Math. 103 (1960), 1-23. MR 0121713, https://doi.org/10.1007/BF02546523
  • [17] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
  • [18] Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326
  • [19] A. M. Vershik and P. V. Sporyshev, Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem, Selecta Math. Soviet. 11 (1992), no. 2, 181-201. Selected translations. MR 1166627
  • [20] V. H. Vu, Sharp concentration of random polytopes, Geom. Funct. Anal. 15 (2005), no. 6, 1284-1318. MR 2221249, https://doi.org/10.1007/s00039-005-0541-8

References

[1]
Fernando Affentranger and Rolf Schneider, Random projections of regular simplices, Discrete Comput. Geom. 7 (1992), no. 3, 219-226. MR 1149653, https://doi.org/10.1007/BF02187839
[2]
Yuliy M. Baryshnikov and Richard A. Vitale, Regular simplices and Gaussian samples, Discrete Comput. Geom. 11 (1994), no. 2, 141-147. MR 1254086, https://doi.org/10.1007/BF02574000
[3]
Mareen Beermann, Random Polytopes, Ph.D. thesis, University of Osnabrück, 2015, Available at: .
[4]
Mareen Beermann and Matthias Reitzner, Monotonicity of functionals of random polytopes, In: G. Ambrus, K. Böröczky, and Z. Füredi, (eds.): Discrete Geometry and Convexity, pp. 23-28, A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, 2017, preprint available at http://arxiv.org/abs/1706.08342.
[5]
Marcel Berger, Geometry. II, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882916
[6]
Ulrich Betke and Martin Henk, Intrinsic volumes and lattice points of crosspolytopes, Monatsh. Math. 115 (1993), no. 1-2, 27-33. MR 1223242, https://doi.org/10.1007/BF01311208
[7]
Johannes Böhm and Eike Hertel, Polyedergeometrie in -dimensionalen Räumen konstanter Krümmung, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series], vol. 70, Birkhäuser Verlag, Basel-Boston, Mass., 1981 (German). MR 626823
[8]
Gilles Bonnet, Julian Grote, Daniel Temesvari, Christoph Thäle, Nicola Turchi, and Florian Wespi, Monotonicity of facet numbers of random convex hulls, J. Math. Anal. Appl. 455 (2017), no. 2, 1351-1364. MR 3671230, https://doi.org/10.1016/j.jmaa.2017.06.054
[9]
Károly Böröczky Jr. and Martin Henk, Random projections of regular polytopes, Arch. Math. (Basel) 73 (1999), no. 6, 465-473. MR 1725183, https://doi.org/10.1007/s000130050424
[10]
Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, and Matthias Reitzner, The monotonicity of -vectors of random polytopes, Electron. Commun. Probab. 18 (2013), no. 23, 8. MR 3044471, https://doi.org/10.1214/ECP.v18-2469
[11]
David L. Donoho and Jared Tanner, Counting faces of randomly projected polytopes when the projection radically lowers dimension, J. Amer. Math. Soc. 22 (2009), no. 1, 1-53. MR 2449053, https://doi.org/10.1090/S0894-0347-08-00600-0
[12]
David L. Donoho and Jared Tanner, Counting the faces of randomly-projected hypercubes and orthants, with applications, Discrete Comput. Geom. 43 (2010), no. 3, 522-541. MR 2587835, https://doi.org/10.1007/s00454-009-9221-z
[13]
Branko Grünbaum, Convex polytopes, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR 1976856
[14]
Daniel Hug, Götz Olaf Munsonius, and Matthias Reitzner, Asymptotic mean values of Gaussian polytopes, Beiträge Algebra Geom. 45 (2004), no. 2, 531-548. MR 2093024
[15]
K. Leichtweiss, Konvexe Mengen, Hochschulbücher für Mathematik [University Books for Mathematics], 81, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980 (German). MR 559138
[16]
Harold Ruben, On the geometrical moments of skew-regular simplices in hyperspherical space, with some applications in geometry and mathematical statistics, Acta Math. 103 (1960), 1-23. MR 0121713, https://doi.org/10.1007/BF02546523
[17]
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
[18]
Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326
[19]
A. M. Vershik and P. V. Sporyshev, Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem, Selecta Math. Soviet. 11 (1992), no. 2, 181-201. Selected translations. MR 1166627
[20]
V. H. Vu, Sharp concentration of random polytopes, Geom. Funct. Anal. 15 (2005), no. 6, 1284-1318. MR 2221249, https://doi.org/10.1007/s00039-005-0541-8

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Additional Information

Zakhar Kabluchko
Affiliation: Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Orléans-Ring 10, 48149 Münster, Germany
Email: zakhar.kabluchko@uni-muenster.de

Christoph Thäle
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
Email: christoph.thaele@rub.de

DOI: https://doi.org/10.1090/proc/13827
Keywords: Convex hull, Gaussian polytope, Gaussian zonotope, Goodman--Pollack model, $f$-vector, random polytope, regular polytope
Received by editor(s): April 26, 2017
Published electronically: October 6, 2017
Communicated by: David Levin
Article copyright: © Copyright 2017 American Mathematical Society

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