Abstract
We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O d (m 2/3 k 2/3 n (d−2)/3+kn d−2+m) incidences between the k red points and m hyperplanes spanned by all n points provided that m=Ω(n d−2). For the monochromatic case k=n, this was proved by Agarwal and Aronov (Discrete Comput. Geom. 7(4):359–369, 1992).
We use this incidence bound to prove that a set of n points, no more than n−k of which lie on any plane or two lines, spans Ω(nk 2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy’s (Erdős and Purdy in Handbook of Combinatorics, vol. 1, pp. 809–873, Elsevier, Amsterdam, 1995) on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.
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Agarwal, P.K., Aronov, B.: Counting facets and incidences. Discrete Comput. Geom. 7(4), 359–369 (1992)
Beck, J.: On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry. Combinatorica 3(3), 281–297 (1983)
Brass, P., Knauer, C.: On counting point-hyperplane incidences. Comput. Geom., Theory Appl. 25(1–2), 13–20 (2003)
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, Berlin (2005)
Dirac, G.A.: Collinearity properties of sets of points. Q. J. Math. 2(1), 221–227 (1951)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)
Edelsbrunner, H., Sharir, M.: A hyperplane incidence problem with applications to counting distances. In: SIGAL International Symposium on Algorithms, pp. 419–428 (1990)
Edelsbrunner, H., Guibas, L., Sharir, M.: The complexity of many cells in arrangements of planes and related problems. Discrete Comput. Geom. 5(1), 197–216 (1990)
Elekes, G., Tóth, C.D.: Incidences of not-too-degenerate hyperplanes. In: SCG ’05: Proceedings of the Twenty-First Annual Symposium on Computational Geometry, pp. 16–21. ACM, New York (2005)
Erdős, P.: On the combinatorial problems which I would most like to see solved. Combinatorica 1, 24–42 (1981)
Erdős, P.: On some problems of elementary and combinatorial geometry. Ann. Mat. Appl., Ser. IV CIII, 99–108
Erdős, P., Purdy, G.: Extremal problems in combinatorial geometry. In: Graham, R., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. 1, pp. 809–873. Elsevier, Amsterdam (1995)
Grünbaum, B., Shephard, G.: Simplicial arrangements in projective 3-space. Mitt. Math., Semin. Giessen 166, 49–101 (1984)
Hansen, S.: A generalization of a theorem of Sylvester on the lines determined by a finite point set. Math. Scand. 16, 175–180 (1965)
Motzkin, T.: The lines and planes connecting the points of a finite set. Trans. Am. Math. Soc. 70(3), 451–464 (1951)
Pach, J., Sharir, M.: Geometric incidences. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, pp. 185–223. American Mathematical Society, Providence (2004)
Purdy, G.: A proof of a consequence of Dirac’s conjecture. Geom. Dedic. 10, 317–321 (1981)
Purdy, G.: Two results about points, lines and planes. Discrete Math. 60, 215–218 (1986)
Szemerédi, E., Trotter, J.W.T.: Extremal problems in discrete geometry. Combinatorica 3(3–4), 381–392 (1983)
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Lund, B.D., Purdy, G.B. & Smith, J.W. A Bichromatic Incidence Bound and an Application. Discrete Comput Geom 46, 611–625 (2011). https://doi.org/10.1007/s00454-011-9367-3
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DOI: https://doi.org/10.1007/s00454-011-9367-3