Pushing disks apart - The Kneser-Poulsen conjecture in the plane

We give a proof of the planar case of a longstanding conjecture of Kneser (1955) and Poulsen (1954). In fact, we prove more by showing that 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.

[1]  Balázs Csikós.  On the Volume of the Union of Balls , 1998, Discret. Comput. Geom..

[2]  Vasilis Capoyleas.  On the Area of the Intersection of Disks in the Plane , 1996, Comput. Geom..

[3]  M. D. Kirszbraun.  Über die zusammenziehende und Lipschitzsche Transformationen , 1934 .

[4]  M. Kneser,et al.  Einige Bemerkungen über das Minkowskische Flächenmaß , 1955 .

[5]  阿部 浩一,et al.  Fundamenta Mathematicae私抄 : 退任の辞に代えて , 1987 .

[6]  M. Meyer,et al.  The volume of the intersection of a convex body with its translates , 1992, math/9204212.

[7]  Herbert Edelsbrunner,et al.  The union of balls and its dual shape , 1993, SCG '93.

[8]  R. Alexander.  The Circumdisk and its Relation to a Theorem of Kirszbraun and Valentine , 1984 .

[9]  János Pach,et al.  On the Perimeter of a Point Set in the Plane , 1990, Discrete and Computational Geometry.

[10]  M. Gromov,et al.  Monotonicity of the volume of intersection of balls , 1987 .

[11]  Victor Klee,et al.  Old And New Unsolved Problems In Plane Geometry And Number Theory , 1991 .

[12]  V. Klee.  Some Unsolved Problems in Plane Geometry , 1979 .

[13]  Amit Sahai,et al.  Pushing disks together—the continuous-motion case , 1996, STOC '96.

[14]  W. Rehder.  On the Volume of Unions of Translates of a Convex Set , 1980 .

[15]  R. Alexander.  Lipschitzian mappings and total mean curvature of polyhedral surfaces. I , 1985 .

[16]  Raimund Seidel,et al.  Exact Upper Bounds for the Number of Faces in d-Dimensional Voronoi Diagrams , 1990, Applied Geometry And Discrete Mathematics.

[17]  Kenneth Falconer,et al.  Unsolved Problems In Geometry , 1991 .

[18]  Alan Shuchat.  Matrix and Network Models in Archaeology , 1984 .