Abstract
We consider the following problem. Let n ≥ 2, b ≥ 1 and q ≥ 2 be integers. Let R and B be two disjoint sets of n red points and bn blue points in the plane, respectively, such that no three points of R∪B lie on the same line. Let n = n 1 + n 2 + ... + n q be an integer-partition of n such that 1 ≤ n i for every 1 ≤ i ≤ q. Then we want to partition R ∪ B into q disjoint subsets P 1 ∪ P 2 ∪ ... ∪ P q that satisfy the following two conditions: (i) conv (P i ) ∩ conv (P j ) = ⊘ for all 1 ≤ i < j ≤ q, where conv(P i ) denotes the convex hull of P i ; and (ii) each P i contains exactly n i red points and bn i blue points for every 1 ≤ i ≤ q.
We shall prove that the above partition exists in the case where (i) 2 ≤ n ≤ 8 and 1 ≤ n i ≤ n/2 for every 1 ≤ i ≤ q, and (ii) n 1 = n 2 = ... = n q-1 = 2 and n q =1.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
J. Akiyma, A. Kaneko, M. Kano, G. Nakamura, E. Rivera-Campo, S. Tokunaga and J. Urrutia, Radical Perfect partitions of convex sets in the plane. Discrete and Computational Geometry (Lecture notes in computer science No.1763) 1–13, (2000).
S. Bespamyatnikh, D. Kirkpatrick and J. Snoeyink, Generalizing ham sandwich cuts to equitable subdivisions, preprint.
J. Goodman and J. O’Rourke, Handbook of Discrete and Computational Geometry, CRC Press, p.211 (1997)
H. Ito, H. Uehara, and M. Yokoyama, 2-dimentinal ham-sandwich theorem for partitioning into three convex pieces, Discrete and Computational Geometry (Lecture notes in computer science No.1763), 129–157, (2000).
A. Kaneko, and M. Kano, Balanced partitions of two sets of points in the plane. Computational Geometry: Theory and Applications, 13, 253–261 (1999).
A. Kaneko, and M. Kano, A balanced partition of points in the plane and tree embedding problems, preprint Computatinal Geometry: Theory and Applications, 13, 253–261 (1999).
A. Kaneko, and M. Kano, Perfect partitions of convex sets in the plane, preprint.
T. Sakai, Balanced Convex Partitions of Measures in R 2, to appear in Graphs and Combinatorics.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kaneko, A., Kano, M. (2001). Generalized Balanced Partitions of Two Sets of Points in the Plane. In: Akiyama, J., Kano, M., Urabe, M. (eds) Discrete and Computational Geometry. JCDCG 2000. Lecture Notes in Computer Science, vol 2098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47738-1_16
Download citation
DOI: https://doi.org/10.1007/3-540-47738-1_16
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42306-5
Online ISBN: 978-3-540-47738-9
eBook Packages: Springer Book Archive