Monotonicity of expected 𝑓-vectors for projections of regular polytopes

Kabluchko, Zakhar; Thäle, Christoph

doi:10.1090/proc/13827

Monotonicity of expected $f$-vectors for projections of regular polytopes
HTML articles powered by AMS MathViewer

by Zakhar Kabluchko and Christoph Thäle

Proc. Amer. Math. Soc. 146 (2018), 1295-1303

DOI: https://doi.org/10.1090/proc/13827

Published electronically: October 6, 2017

PDF | Request permission

Abstract:

Let $P_n$ be an $n$-dimensional regular polytope from one of the three infinite series (regular simplices, regular crosspolytopes, and cubes). Project $P_n$ onto a random, uniformly distributed linear subspace of dimension $d\geq 2$. We prove that the expected number of $k$-dimensional faces of the resulting random polytope is an increasing function of $n$. As a corollary, we show that the expected number of $k$-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.

References

Fernando Affentranger and Rolf Schneider, Random projections of regular simplices, Discrete Comput. Geom. 7 (1992), no. 3, 219–226. MR 1149653, DOI 10.1007/BF02187839
Yuliy M. Baryshnikov and Richard A. Vitale, Regular simplices and Gaussian samples, Discrete Comput. Geom. 11 (1994), no. 2, 141–147. MR 1254086, DOI 10.1007/BF02574000
Mareen Beermann, Random Polytopes, Ph.D. thesis, University of Osnabrück, 2015, Available at: https://repositorium.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2015062313276.
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.
Marcel Berger, Geometry. II, Universitext, Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy. MR 882916, DOI 10.1007/978-3-540-93816-3
Ulrich Betke and Martin Henk, Intrinsic volumes and lattice points of crosspolytopes, Monatsh. Math. 115 (1993), no. 1-2, 27–33. MR 1223242, DOI 10.1007/BF01311208
Johannes Böhm and Eike Hertel, Polyedergeometrie in $n$-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
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, DOI 10.1016/j.jmaa.2017.06.054
Károly Böröczky Jr. and Martin Henk, Random projections of regular polytopes, Arch. Math. (Basel) 73 (1999), no. 6, 465–473. MR 1725183, DOI 10.1007/s000130050424
Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, and Matthias Reitzner, The monotonicity of $f$-vectors of random polytopes, Electron. Commun. Probab. 18 (2013), no. 23, 8. MR 3044471, DOI 10.1214/ECP.v18-2469
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, DOI 10.1090/S0894-0347-08-00600-0
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, DOI 10.1007/s00454-009-9221-z
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, DOI 10.1007/978-1-4613-0019-9
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
K. Leichtweiss, Konvexe Mengen, Hochschulbücher für Mathematik [University Books for Mathematics], vol. 81, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980 (German). MR 559138, DOI 10.1007/978-3-642-95335-4
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 121713, DOI 10.1007/BF02546523
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
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
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
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