Abstract
An interior point of a finite point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let g(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least g(k) interior points has a subset of points containing exactly k interior points. Similarly, for any integer k ≥ 3, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing exactly k or k+1 interior points. In this note, we show that g(k) ≥ 3k-1 for k ≥ 3. We also show that h(k) ≥ 2k+1 for 5 ≤ k ≤ 8, and h(k) ≥ 3k-7 for k ≥ 8.
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References
Avis, D., Hosono, K., Urabe, M.: On the existance of a point subset with 4 or 5 interior points. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 57–64. Springer, Heidelberg (2000)
Avis, D., Hosono, K., Urabe, M.: On the existence of a point subset with a specified number of interior points. Discrete Mathematics 241(1-3), 33–40 (2001)
Erdös, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Mathematica 2, 463–470 (1935)
Horton, J.: Sets with no empty 7-gons. Canad. Math. Bull. 26, 482–484 (1983)
Hosono, K., Károlyi, G., Urabe, M.: Constructions from empty polygons. In: Bezdek, A. (ed.) Discrete Geometry: in Honor of W. Kuperberg’s 60th Birthday, pp. 351–358. Marcel Dekker, New York (2003)
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Fevens, T. (2003). A Note on Point Subsets with a Specified Number of Interior Points. In: Akiyama, J., Kano, M. (eds) Discrete and Computational Geometry. JCDCG 2002. Lecture Notes in Computer Science, vol 2866. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44400-8_15
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DOI: https://doi.org/10.1007/978-3-540-44400-8_15
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