Problem 18: Pushing Disks Together

Statement

When a collection of disks are pushed closer together, so that no distance between two center points increases, can the area of their union increase?

Origin

Kneser (1955) and Poulsen (1954).

Status/Conjectures

Solved by K. Bezdek and R. Connelly. See their web page. (Update as of 3 Aug. 2000.)

Partial and Related Results

Previously only settled in the continuous-motion case [BS98], for both this and the corresponding question for intersection area decrease [Cap96]. But now both solved; see above.

Appearances

[MO01]

Categories

combinatorial geometry

Entry Revision History

J. O'Rourke, 2 Aug. 2001; 3 Aug. 2003.

Bibliography

BS98

Marshall Bern and Amit Sahai.
Pushing disks together - The continuous-motion case.
Discrete Comput. Geom., 20:499-514, 1998.

Cap96

V. Capoyleas.
On the area of the intersection of disks in the plane.
Comput. Geom. Theory Appl., 6:393-396, 1996.

MO01

J. S. B. Mitchell and Joseph O'Rourke.
Computational geometry column 42.
Internat. J. Comput. Geom. Appl., 11(5):573-582, 2001.
Also in SIGACT News 32(3):63-72 (2001), Issue 120.