Abstract
We show by a construction that there are at least exp {cV (d−1)/(d+1)} convex lattice polytopes in ℝd of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation.
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References
G. E. Andrews, A lower bound for the volumes of strictly convex bodies with many boundary points, Trans. Amer. Math. Soc., 106 (1993), 270–279.
V. I. Arnol’d, Statistics of integral convex polytopes, Funk. Anal. Pril., 14 (1980), 1–3 (in Russian). English translation, Funct. Anal. Appl., 14 (1980), 79–84.
I. Bárány, Random points and lattice points in convex bodies, Bull. Amer. Math. Soc., 45 (2008), 339–365.
I. Bárány and D. G. Larman, The convex hull of the integer points in a large ball, Math. Annalen, 312 (1998), 167–181.
I. Bárány and J. Pach, On the number of convex lattice polygons, Combinatorics, Probability, and Computation, 1 (1992), 295–302.
I. Bárány and A. Vershik, On the number of convex lattice polytopes, 24 (2004), 171–185.
S. Konyagin and S. V. Sevastyanov, Estimation of the number of vertices of a convex integral polyhedron in terms of its volume, Funk. Anal. Pril., 18 (1984), 13–15 (in Russian). English translation, Funct. Anal. Appl., 18 (1984), 11–13.
Sh. Reizner, C. Schütt and E. Werner, Dropping a vertex or a facet from a convex polytope, Forum Mathematicum, 13 (2001), 359–378.
C. Zong, On Arnold’s problem on the classifications of convex lattice polytopes, manuscript (2011), arXiv:1107.2966.
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Bárány, I. On a question of V. I. Arnol’d. Acta Math Hung 137, 72–81 (2012). https://doi.org/10.1007/s10474-012-0219-2
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DOI: https://doi.org/10.1007/s10474-012-0219-2