Geometriae Dedicata

, Volume 162, Issue 1, pp 95–107

# Strict Kneser–Poulsen conjecture for large radii

Article

## Abstract

In this paper we prove the Kneser–Poulsen conjecture for the case of large radii. Namely, if a finite number of points in Euclidean space $${\mathbb{E}^n}$$ is rearranged so that the distance between each pair of points does not decrease, then there exists a positive number r 0 that depends on the rearrangement of the points, such that if we consider n-dimensional balls of radius rr 0 with centers at these points, then the volume of the union (intersection) of the balls before the rearrangement is not less (not greater) than the volume of the union (intersection) after the rearrangement. Moreover, the inequality is strict whenever the new point set is not congruent to the original one. Also under the same conditions we prove a similar result about surface volumes instead of volumes. In order to prove the above mentioned results we use ideas from tensegrity theory to strengthen the theorem of Sudakov (Dokl. Akad. Nauk SSSR 197:43–45, 1971), Alexander (Trans. Am. Math. Soc., 288(2):661–678, 1985) and Capoyleas and Pach (Discrete and computational geometry. American Mathematical Society, Providence, 1991), which says that the mean width of the convex hull of a finite number of points does not decrease after an expansive rearrangement of those points. In this paper we show that the mean width increases strictly, unless the expansive rearrangement was a congruence. We also show that if the configuration of centers of the balls is fixed and the volume of the intersection of the balls is considered as a function of the radius r, then the second highest term in the asymptotic expansion of this function is equal to $${-M_nr^{n-1}}$$ , where M n is the mean width of the convex hall of the centers. This theorem was conjectured by Balázs Csikós in 2009.

## Keywords

Kneser–Poulsen conjecture Volume inequalities Perimeter inequalities

## Mathematics Subject Classification

52A40 52A20 52A38

## References

1. 1.
Alexander R.: Lipschitzian mappings and total mean curvature of polyhedral surfaces. I. Trans. Am. Math. Soc. 288(2), 661–678 (1985)
2. 2.
Bezdek K., Connelly R.: Pushing disks apart—the Kneser–Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236 (2002)
3. 3.
Bezdek K., Connelly R.: The Kneser–Poulsen conjecture for spherical polytopes. Discret. Comput. Geom. 32(1), 101–106 (2004)
4. 4.
Bezdek K., Connelly R., Csikós B.: On the perimeter of the intersection of congruent disks. Beiträge Algebra Geom. 47(1), 53–62 (2006)
5. 5.
Capoyleas, V., Pach, J.: On the perimeter of a point set in the plane. In: Discrete and Computational Geometry (New Brunswick, 1989/1990), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 6, pp. 67–76. American Mathematical Society, Providence (1991)Google Scholar
6. 6.
Connelly R.: Rigidity and energy. Invent. Math. 66(1), 11–33 (1982)
7. 7.
Csikós, B.: On the Hadwiger–Kneser–Poulsen conjecture. In: Intuitive Geometry (Budapest 1995), Bolyai Society Mathematical Studies, vol. 6, pp. 291–299. János Bolyai Mathematical Society, Budapest (1997)Google Scholar
8. 8.
Csikós B.: On the volume of the union of balls. Discret. Comput. Geom. 20(4), 449–461 (1998)
9. 9.
Csikós B.: On the volume of flowers in space forms. Geom. Dedic. 86(1–3), 59–79 (2001)
10. 10.
Klee, V., Wagon, S.: Old and New Unsolved Problems in Plane Geometry and Number Theory, The Dolciani Mathematical Expositions, vol. 11. Mathematical Association of America, Washington (1991)Google Scholar
11. 11.
Kneser M.: Einige Bemerkungen über das Minkowskische Flächenmass. Arch. Math. (Basel) 6, 382–390 (1955)
12. 12.
Poulsen E.T.: Problem 10. Math. Scand. 2, 346 (1954)Google Scholar
13. 13.
Sudakov V.N.: Gaussian random processes, and measures of solid angles in Hilbert space. Dokl. Akad. Nauk SSSR 197, 43–45 (1971)
14. 14.
Whiteley W.: Infinitesimally rigid polyhedra. I. Statics of frameworks. Trans. Am. Math. Soc. 285(2), 431–465 (1984)