If a finite set of disks in the plane is rearranged so that the
distance between each pair of centers does not decrease, then the area of
the union does not decrease, and the area of the intersection does not
increase. This very basic geometric property of the Euclidean plane was
conjectured by Kneser and Poulsen in the 1950's and described in Chapter 3
of Klee and Wagon's book on unsolved problems in plane geometry and number
theory. The proof with Károly Bezdek not only provided a solution
to the problem in the plane, but also introduced at least three new
techniques for this and related Kneser-Poulsen type problems. There are
several related problems and extensions to higher dimensions that are
still open, including the original Kneser-Poulsen problem for dimensions
greater than 2.
R E F R E S H M E N T S
4:00 To 4:25 PM
Kenneth W. Anderson
Memorial Reading Room
