Pushing things around

  • Date: 10/12/2007

Robert Connelly (Cornell University)


University of Calgary


Unsolved for over twenty-five years, a surprisingly difficult
conjecture stated that a non-crossing polygonal chain of fixed-length
edges in the plane can be continuously opened without crossing. Gunter
Rote, Erik Demaine and I proved this Carpenter's Rule conjecture in

Unsolved for almost fifty years, M. Kneser and E. M. Poulsen
conjectured that if a finite collection of circular disks in the plane
are picked up and re-positioned in the plane so that no pair of centers
gets closer together, then the area of the union never gets smaller.
Karoly Bezdek and I proved this Kneser-Poulsen conjecture in 2002.

These two seemingly disparate results, the proofs of the Carpenter's
Rule conjecture and the Kneser-Poulsen conjecture, can be combined to
extend the Carpenter's Rule result to chains of appropriately slender
sets, which can open without self-intersection. One of the critical
ingredients in the Kneser-Poulsen result is a formula by Balazs Csikos
for the change in the area of unions and intersections of disks as the
centers are moved continuously. This brings up a large class of
problems simultaneously generalizing the notions of packings and
coverings of disks. For example, it shows how to find the critical area
configurations of the union of one set of disks, with one larger
radius, intersected with the union of another set of disks, with
another smaller radius.

Other Information: 

PIMS Distinguished Lecture 2007