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The Kneser–Poulsen Conjecture for Special Contractions

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Abstract

The Kneser–Poulsen conjecture states that if the centers of a family of N unit balls in \({\mathbb E}^d\) are contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions. First, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that \(N\ge (1+\sqrt{2})^d\). Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union. Second, a strong contraction is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.

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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.: Lectures on Sphere Arrangements: The Discrete Geometric Side. Fields Institute Monographs, vol. 32. Springer, New York (2013)

  3. 3.

    Bezdek, K., Connelly, R.: Pushing disks apart: the Kneser–Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236 (2002)

  4. 4.

    Bezdek, K., Lángi, Z.: Density bounds for outer parallel domains of unit ball packings. Proc. Steklov Inst. Math. 288(1), 209–225 (2015)

  5. 5.

    Bezdek, K., Lángi, Z., Naszódi, M., Papez, P.: Ball-polyhedra. Discrete Comput. Geom. 38(2), 201–230 (2007)

  6. 6.

    Böröczky Jr., K., Wintsche, G.: Covering the sphere by equal spherical balls. In: Aronov, B., et al. (eds.) Discrete and Computational Geometry. Algorithms and Combinatorics, vol. 25, pp. 235–251. Springer, Berlin (2003)

  7. 7.

    Bouligand, G.: Ensembles impropres et nombre dimensionnel. Bull. Sci. Math. 52, 320–344 (1928)

  8. 8.

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

  9. 9.

    Csikós, B.: On the volume of the union of balls. Discrete Comput. Geom. 20(4), 449–461 (1998)

  10. 10.

    Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Proceedings of Symposia in Pure Mathematics, vol. VII, pp. 101–180. American Mathematical Society, Providence (1963)

  11. 11.

    Fodor, F., Kurusa, Á., Vígh, V.: Inequalities for hyperconvex sets. Adv. Geom. 16(3), 337–348 (2016)

  12. 12.

    Gardner, R.J.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39(3), 355–405 (2002)

  13. 13.

    Gromov, M.: Monotonicity of the volume of intersection of balls. In: Lindenstrauss, J., Milman, V.D. (eds.) Geometrical Aspects of Functional Analysis (1985/86). Lecture Notes in Mathematics, vol. 1267, pp. 1–4. Springer, Berlin (1987)

  14. 14.

    Jung, H.: Über die kleinste Kugel, die eine räumliche Figur einschliesst. J. Reine Angew. Math. 123, 241–257 (1901)

  15. 15.

    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, DC (1991)

  16. 16.

    Kneser, M.: Einige Bemerkungen über das Minkowskische Flächenmass. Arch. Math. 6, 382–390 (1955)

  17. 17.

    Paouris, G., Pivovarov, P.: Random ball-polyhedra and inequalities for intrinsic volumes. Monatsh. Math. 182(3), 709–729 (2017)

  18. 18.

    Poulsen, E.T.: Problem 10. Math. Scand. 2, 346 (1954)

  19. 19.

    Rehder, W.: On the volume of unions of translates of a convex set. Am. Math. Mon. 87(5), 382–384 (1980)

  20. 20.

    Rogers, C.A.: Packing and Covering. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 54. Cambridge University Press, New York (1964)

  21. 21.

    Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 151, 2nd edn. Cambridge University Press, Cambridge (2014)

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Acknowledgements

We thank Peter Pivovarov and Ferenc Fodor for our discussions. Károly Bezdek was partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. Márton Naszódi was partially supported by the National Research, Development and Innovation Office (NKFIH) grants: NKFI-K119670 and NKFI-PD104744 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, as well as the ÚNKP-17-4 New National Excellence Program of the Ministry of Human Capacities. Part of his research was carried out during a stay at EPFL, Lausanne at János Pach’s Chair of Discrete and Computational Geometry supported by the Swiss National Science Foundation Grants 200020-162884 and 200021-165977.

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Correspondence to Márton Naszódi.

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Editor in Charge: János Pach

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Bezdek, K., Naszódi, M. The Kneser–Poulsen Conjecture for Special Contractions. Discrete Comput Geom 60, 967–980 (2018). https://doi.org/10.1007/s00454-018-9976-1

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Keywords

  • Kneser–Poulsen conjecture
  • Alexander’s contraction
  • Ball-polyhedra
  • Volume of intersections of balls
  • Volume of unions of balls
  • Blaschke–Santalo inequality

Mathematics Subject Classification

  • 52A20
  • 52A22