Abstract
In this article we consider S to be a set of points in d-space with the property that any d points of S span a hyperplane and not all the points of S are contained in a hyperplane. The aim of this article is to introduce the function \(e_d(n)\), which denotes the minimal number of hyperplanes meeting S in precisely d points, minimising over all such sets of points S with \(|S|=n\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ball, S.: On sets defining few ordinary planes, arXiv:1606.02138, (2016)
Borwein, P.B., Moser, W.O.J.: A survey of Sylvester’s problem and its generalizations. Aequationes Math. 40, 111–135 (1990)
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York, p. 500 (2005)
de Bruijn, N.G., Erdős, P.: On a combinatorial problem. Neder. Akad. Wetensch Proc. 51, 1277–1279 (1948)
Coxeter, H.S.M.: Projective Geometry, 2nd edn. Springer, New York, pp. 213–313 (2008)
Crowe, D.W., McKee, T.A.: Sylvester’s problem on collinear points. Math. Mag. 41, 30–34 (1968)
Csima, J., Sawyer, E.: There exist \(6n/13\) ordinary points. Discrete Comput. Geom. 9, 187–202 (1993)
Dirac, G.A.: Collinearity properties of sets of points. Quart. J. Math. 2, 221–227 (1951)
Gallai, T.: Solution to Problem 4065. Amer. Math. Monthly 51, 169–171 (1944)
Green, B., Tao, T.: On sets defining few ordinary lines. Discrete Comput. Geom. 50, 409–468 (2013)
Kelly, L.M., Moser, W.: On the number of ordinary lines determined by \(n\) points. Canad. J. Math. 10, 210–219 (1958)
Monserrat, J.: Generalization of Sylvester problem, Bachelors Degree Thesis, Universitat Politècnica Catalunya, Barcelona, (2015)
Moser, W.: Abstract Groups and Geometrical Configurations, Ph.D. Thesis, Univ. of Toronto (1957)
Motzkin, T.: The lines and planes connecting the points of a finite set. Trans. Amer. Math. Soc. 70, 451–464 (1951)
Sylvester, J.J.: Mathematical question 11851. Educational Times 59, 98 (1893)
Author information
Authors and Affiliations
Corresponding author
Additional information
S. Ball acknowledges the support of the project MTM2014-54745-P of the Spanish Ministerio de Economía y Competitividad.
