1. CMPS 3130/6130 Computational Geometry Spring 2017
Ham Sandwich Theorem Carola Wenk
4/4/17
CMPS 3130/6130 Computational Geometry

2. CMPS 3130/6130 Computational Geometry
Ham-Sandwich Theorem
Theorem: Let P and Q be two finite point sets in the plane. Then there exists a line l such that on each side of l there are at most |P|/2 points of P and at most |Q|/2 points of Q.
4/4/17
CMPS 3130/6130 Computational Geometry

3. CMPS 3130/6130 Computational Geometry
Ham-Sandwich Theorem
Proof:
Find a line l such that on each side of l there are at most |P|/2 points of P.
Then rotate l counter-clockwise in such a way that there are always at most |P|/2 points of P on each side of l.
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CMPS 3130/6130 Computational Geometry

4. CMPS 3130/6130 Computational Geometry
Rotation
Left: 4 Right: 4
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CMPS 3130/6130 Computational Geometry

5. Rotation Left: 4 Right: 3 Rotate around this point now 4/4/17
CMPS 3130/6130 Computational Geometry

6. CMPS 3130/6130 Computational Geometry
Rotation
Left: 4 Right: 4
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CMPS 3130/6130 Computational Geometry

7. CMPS 3130/6130 Computational Geometry
Rotation
Left: 3 Right: 4
rotate
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CMPS 3130/6130 Computational Geometry

8. CMPS 3130/6130 Computational Geometry
Rotation
Left: 3 Right: 4
rotate
4/4/17
CMPS 3130/6130 Computational Geometry

9. CMPS 3130/6130 Computational Geometry
Rotation
Left: 4 Right: 3
rotate
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CMPS 3130/6130 Computational Geometry

10. CMPS 3130/6130 Computational Geometry
Rotation
Left: 3 Right: 4
rotate
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CMPS 3130/6130 Computational Geometry

11. CMPS 3130/6130 Computational Geometry
Rotation
Left: 4 Right: 3
rotate
4/4/17
CMPS 3130/6130 Computational Geometry

12. Rotation Left: 2 Right: 4 Rotate around this point now 4/4/17
CMPS 3130/6130 Computational Geometry

13. Rotation Left: 4 Right: 4 Rotate around this point now 4/4/17
CMPS 3130/6130 Computational Geometry

14. CMPS 3130/6130 Computational Geometry
Rotation
Left: 4 Right: 3
rotate
4/4/17
CMPS 3130/6130 Computational Geometry

15. CMPS 3130/6130 Computational Geometry
Rotation
Left: 3 Right: 4
rotate
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CMPS 3130/6130 Computational Geometry

16. CMPS 3130/6130 Computational Geometry
Rotation
Left: 4 Right: 3
rotate
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CMPS 3130/6130 Computational Geometry

17. CMPS 3130/6130 Computational Geometry
Rotation
Left: 4 Right: 4
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CMPS 3130/6130 Computational Geometry

18. CMPS 3130/6130 Computational Geometry
Proof Continued
In general, choose the rotation point such that the number of points on each side of l does not change.
k points
m points
rotate
rotate
m points
k points
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CMPS 3130/6130 Computational Geometry

19. CMPS 3130/6130 Computational Geometry
Proof Continued
Throughout the rotation, there are at most |P|/2 points on each side of l.
After 180 rotation, we get the line which is l but directed in the other direction.
Let t be the number of blue points to the left of l at the beginning. In the end, t points are on the right side of l, so |Q|-t-1 are on the left side. Therefore, there must have been one orientation of l such that there were t most |Q|/2 points on each side of l.
4/4/17
CMPS 3130/6130 Computational Geometry