2004 Fiscal Year Final Research Report Summary
BALANCED PARTITIONS OF TWO SETS OF POINTS IN THE PLANE
| Project/Area Number |
15540137
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | IBARAKI UNIVERSITY (2004)
Kogakuin University (2003) |
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Principal Investigator |
KANO Mikio IBARAKI UNIVERSITY, FACULTY OF ENGINEERING, PROFESSOR, 工学部, 教授 (20099823)
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| Project Period (FY) |
2003 – 2004
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| Keywords | Balanced Partition / Balanced subdivision / Red and blue points / Geometric graph / Discrete geometry |
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| Research Abstract |
Recently, the Ham-sandwich Theorem was generalized as follows : If|R|=ag and|B|=bg, then there exists a subdivision X_1∪X_2∪・・・∪X_g of the plane into g disjoint convex polygons such that every X_i contains exactly a red points and b blue points. This theorem was proved by A. Kano and M.Kano for a=1,2 and they proposed it as a conjecture, and then this conjecture was independently proved by three papers.
We obtain the following three new results.
(1)Suppose that $R$ is a disjoint union of R_1 and R_2. Let|R_1|=g_1 and|R_2|=g_2. If|B|=(m-1)g_1+ mg_2, then we can subdivide the plane into g_1+g_2 disjoint convex polygons X_1∪・・・X_{g_1}∪Y_1∪・・・∪Y_{g_2} so that every X_i contains exactly one red point of R_1 and m-1 blue points, and every Y_j contains exactly one red point of R_2 and m blue points.
(2)Let a≧1, g≧0 and h≧0 be integers such that g+h≧1. If|R|=ag+(a+1)h and|B|=(a+1)g+ah, then there exists a subdivision X_1∪・・・∪X_g∪Y_1∪・・・∪ Y_h of the plane into g+h disjoint convex polygons such that every X_i contains exactly a red points and a+1 blue points and every Y_j contains exactly a+1 red points and a blue points.
(3)If|R|=a(g_1+g_2)+(a+1)g_3 and|B|=bg_1+(b+1)(g_2+g_3), then there exists a subdivision X_1∪・・・∪X_{g_1}∪Y_1∪・・・∪ Y_{g_2}∪Z_{1}∪・・・∪Z_{g_3} of the plane into g_1+g_2+g_3 disjoint convex polygons such that every X_i contains exactly a red points and b blue points, every Y_i, if any, contains exactly a red points and b+1 blue points, and every Z_i, if any, contains exactly a+1 red points and b+1 blue points
We also study some problems on discrete geometry and graph theory related to the above problems.
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