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Problem 18: Pushing Disks Together


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?


Kneser (1955) and Poulsen (1954).


Solved by K. Bezdek and R. Connelly. See . (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.




combinatorial geometry

Entry Revision History

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



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


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


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.