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Monotonicity of expected $ f$-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: 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.


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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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