Abstract
Taking up a suggestion of David Gale from 1956, we generate sets of combinatorially isomorphic polytopes by choosing their Gale diagrams at random. We find that in high dimensions, and under suitable assumptions on the growth of the involved parameters, the obtained polytopes have strong neighborliness properties, with high probability.
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Baryshnikov, Y.M., Vitale, R.A.: Regular simplices and Gaussian samples. Discrete Comput. Geom. 11, 141–147 (1994)
Carathéodory, C.: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)
Cover, T.M., Efron, B.: Geometrical probability and random points on a hypersphere. Ann. Math. Stat. 38, 213–220 (1967)
Donoho, D., Tanner, J.: Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102(27), 9452–9457 (2005)
Donoho, D., Tanner, J.: Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Am. Math. Soc. 22, 1–53 (2009)
Donoho, D., Tanner, J.: Counting the faces of randomly-projected hypercubes and orthants, with applications. Discrete Comput. Geom. 43, 522–541 (2010)
Gale, D.: Neighboring vertices on a convex polyhedron. In: Kuhn, H.W., Tucker, A.W. (eds.) Linear Inequalities and Related Systems, Annals of Mathematics Studies, no. 38, pp. 255–263. Princeton University Press, Princeton (1956)
Grünbaum, B.: Convex Polytopes. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. Graduate Texts in Mathematics, vol. 221, 2nd edn. Springer-Verlag, New York (2003)
Hug, D., Schneider, R., Threshold phenomena for random cones. (submitted) arXiv:2004.11473v1
Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer-Verlag, New York (2002)
McMullen, P., Shephard, G.C.: Convex Polytopes and the Upper Bound Conjecture. Prepared in collaboration with J.E. Reeve and A.A. Ball. London Mathematical Society Lecture Note Series, 3. Cambridge University Press, London-New York (1971)
McMullen, P.: Transforms, diagrams and representations. In: Tölke, J., Wills, J.M. (eds.) Contributions to Geometry, Siegen 1978, pp. 92–130. Birkhäuser, Basel (1979)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151, 2nd edn. Cambridge University Press, Cambridge (2014)
Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008)
Shemer, I.: Neighborly polytopes. Israel J. Math. 43, 291–314 (1982)
Shephard, G.C.: Diagrams for positive bases. J. Lond. Math. Soc. 2(4), 165–175 (1971)
Vershik, A.M., Sporyshev, P.V.: Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem. Selecta Math. Soviet. 11(2), 181–201 (1992)
Wendel, J.G.: A problem in geometric probability. Math. Scand. 11, 109–111 (1962)
Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer-Verlag, New York (1995)
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Schneider, R. Random Gale diagrams and neighborly polytopes in high dimensions. Beitr Algebra Geom 62, 641–650 (2021). https://doi.org/10.1007/s13366-020-00526-3
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DOI: https://doi.org/10.1007/s13366-020-00526-3