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The Geometry of Homothetic Covering and Illumination

  • Károly Bezdek
  • Muhammad A. Khan
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)

Abstract

At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two sides of the same coin and give rise to one of the important longstanding open problems in discrete geometry, namely, the Illumination Conjecture. In this paper, we survey the activity in the areas of discrete geometry, computational geometry and geometric analysis motivated by this conjecture. Special care is taken to include the recent advances that are not covered by the existing surveys. We also include some of our recent results related to these problems and describe two new approaches – one conventional and the other computer-assisted – to make progress on the illumination problem. Some open problems and conjectures are also presented.

Keywords

Illumination number Illumination conjecture Covering conjecture Separation conjecture X-ray number X-ray conjecture Illumination parameter Covering parameter Covering index Cylindrical covering parameters \(\epsilon \)-net of convex bodies 

MSC (2010):

52A37 52A40 52C15 52C17 

Notes

Acknowledgements

The first author is partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. The second author is supported by a Vanier Canada Graduate Scholarship (NSERC) and Alberta Innovates Technology Futures (AITF).

References

  1. 1.
    M. Aigner, G.M. Ziegler, Proofs from the Book, 4th edn. (Springer, Berlin, 2010)CrossRefGoogle Scholar
  2. 2.
    A. Akopyan, R. Karasev, F. Petrov, Bang’s problem and symplectic invariants. 1–15 (2014). arXiv:1404.0871v1 [math.MG]
  3. 3.
    S. Artstein-Avidan, O. Raz, Weighted covering numbers of convex sets. Adv. Math. 227, 730–744 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Artstein-Avidan, B.A. Slomka, On weighted covering numbers and the Levi-Hadwiger conjecture. Israel J. Math. 209(1), 125–155 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    K. Ball, The plank problem for symmetric bodies. Invent. Math. 104, 535–543 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    T. Bang, A solution of the "Plank problem". Proc. Am. Math. Soc. 2, 990–993 (1951)MathSciNetzbMATHGoogle Scholar
  7. 7.
    K. Bezdek, Körök Optimális Fedései, Ph.D. thesis, Eötvös Loránd University, Budapest, 1979Google Scholar
  8. 8.
    K. Bezdek, Über einige Kreisüberdeckungen. Beiträge Algebra Geom. 14, 7–13 (1983)zbMATHGoogle Scholar
  9. 9.
    K. Bezdek, The problem of illumination of the boundary of a convex body by affine subspaces. Mathematika 38, 362–375 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    K. Bezdek, Hadwiger’s covering conjecture and its relatives. Am. Math. Monthly 99, 954–956 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    K. Bezdek, Research problem 46. Period. Math. Hungar. 24, 119–121 (1992)MathSciNetCrossRefGoogle Scholar
  12. 12.
    K. Bezdek, G. Kiss, M. Mollard, An illumination problem for zonoids. Israel J. Math. 81, 265–272 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    K. Bezdek, On affine subspaces that illuminate a convex set. Beiträge Alg. Geom. 35, 131–139 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    K. Bezdek, T. Bisztriczky, A proof of Hadwiger’s covering conjecture for dual cyclic polytopes. Geom. Dedicata 68, 29–41 (1997)MathSciNetCrossRefGoogle Scholar
  15. 15.
    K. Bezdek, K. Böroczky, G. Kiss, On the successive illumination parameters of convex bodies. Period. Math. Hungar. 53(1–2), 71–82 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    K. Bezdek, Hadwiger-Levi’s covering problem revisited, in New Trends in Discrete and Computational Geometry, ed. by J. Pach (Springer, New York, 1993), pp. 199–233CrossRefGoogle Scholar
  17. 17.
    K. Bezdek, T. Zamfirescu, A characterization of 3-dimensional convex sets with an infinite X-ray number, in Intuitive Geometry, Colloquia Mathematica Societatis Janos Bolyai, vol. 63 (North-Holland, Amsterdam, 1994) pp. 33–38Google Scholar
  18. 18.
    K. Bezdek, Z. Lángi, M. Naszódi, P. Papez, Ball-polyhedra. Discrete Comput. Geom. 38(2), 201–230 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    K. Bezdek, A.E. Litvak, On the vertex index of convex bodies. Adv. Math. 215(2), 626–641 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    K. Bezdek, A.E. Litvak, Covering convex bodies by cylinders and lattice points by flats. J. Geom. Anal. 19(2), 233–243 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    K. Bezdek, G. Kiss, On the X-ray number of almost smooth convex bodies and of convex bodies of constant width. Can. Math. Bull. 52/3, 342–348 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    K. Bezdek, Classical Topics in Discrete Geometry (CMS Books in Mathematics, Springer, New York, 2010)CrossRefGoogle Scholar
  23. 23.
    K. Bezdek, Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width. Discrete Comput. Geom. 47(2), 275–287 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    K. Bezdek, M.A. Khan, On the covering index of convex bodies. Aequationes Math. 90(5), 879–903 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    K. Bezdek, M.A. Khan, Quantitative covering of convex bodies, in International Workshop on Geometry and Symmetry (GeoSym), Veszprém, Hungary, 29 June–3 July, 2015, http://geosym.mik.uni-pannon.hu/images/presentations/khan.pdfGoogle Scholar
  26. 26.
    T. Bisztriczky, Separation in neighbourly 4-polytopes. Studia Sci. Math. Hungar. 39(3-4), 277–289 (2002)MathSciNetzbMATHGoogle Scholar
  27. 27.
    T. Bisztriczky, F. Fodor, A separation theorem for totally-sewn 4-polytopes. Studia Sci. Math. Hungar. 52(3), 386–422 (2015)MathSciNetzbMATHGoogle Scholar
  28. 28.
    B. Bollobás, P.A. Catalin, P. Erdős, Hadwiger’s conjecture is true for almost every graph. Eur. J. Combin. 1(3), 195–199 (1980)MathSciNetCrossRefGoogle Scholar
  29. 29.
    V. Boltyanski, The problem of illuminating the boundary of a convex body. Izv. Mold. Fil. AN SSSR 76, 77–84 (1960)Google Scholar
  30. 30.
    V. Boltyanski, Helly’s theorem for \(H\)-convex sets. Sov. Math. Dokl. 17(1), 78–81 (1976)zbMATHGoogle Scholar
  31. 31.
    V. Boltyanski, I. Gohberg, Results and Problems in Combinatorial Geometry (Cambridge University Press, Cambridge, 1985)CrossRefGoogle Scholar
  32. 32.
    V. Boltyanski, I. Gohberg, Stories about covering and illuminating of convex bodies. Nieuw Arch. Wisk. 13(1), 1–26 (1995)MathSciNetzbMATHGoogle Scholar
  33. 33.
    V. Boltyanski, H. Martini, Combinatorial geometry of belt bodies. Results Math. 28(3–4), 224–249 (1995)MathSciNetCrossRefGoogle Scholar
  34. 34.
    V. Boltyanski, A solution of the illumination problem for belt bodies. Mat. Zametki 58, 505–511 (1996)MathSciNetGoogle Scholar
  35. 35.
    V. Boltyanski, H. Martini, P.S. Soltan, Excursions into Combinatorial Geometry (Springer, New York, 1997)CrossRefGoogle Scholar
  36. 36.
    V. Boltyanski, Solution of the illumination problem for three dimensional convex bodies. Dokl. Akad. Nauk 375, 298–301 (2000)MathSciNetGoogle Scholar
  37. 37.
    V. Boltyanski, Solution of the illumination problem for bodies with \({{\rm {md}}} M=2\). Discrete Comput. Geom. 26, 527–541 (2001)Google Scholar
  38. 38.
    V. Boltyanski, H. Martini, Covering belt bodies by smaller homothetical copies. Beiträge Alg. Geom. 42, 313–324 (2001)MathSciNetzbMATHGoogle Scholar
  39. 39.
    K. Böröczky Jr., D.G. Larman, S. Sezgin, C.M. Zong, On generalized kissing numbers and blocking numbers. Rendiconti del Circolo Mathematico di Palermo, Serie II(Suppl. 65), 39–57 (2000)MathSciNetzbMATHGoogle Scholar
  40. 40.
    K. Böröczky Jr., G. Wintsche, Covering the sphere by equal spherical balls, in The Goodman-Pollack Festschrift (Springer, Berlin, 2003), pp. 237–253Google Scholar
  41. 41.
    K. Borsuk, Drei Sätze über die \(n\)-dimensionale euklidische Sphäre. Fund. Math. 20, 177–190 (1933)CrossRefGoogle Scholar
  42. 42.
    L. Dalla, D.G. Larman, P. Mani-Levitska, C. Zong, The blocking numbers of convex bodies. Discrete Comput. Geom. 24(2-3), 267–277 (2000)MathSciNetCrossRefGoogle Scholar
  43. 43.
    B.V. Dekster, Each convex body in \({\mathbb{E}}^3\) symmetric about a plane can be illuminated by 8 directions. J. Geom. 69(1-2), 37–50 (2000)MathSciNetCrossRefGoogle Scholar
  44. 44.
    P. Erdős, C.A. Rogers, Covering space with convex bodies. Acta Arith. 7, 281–285 (1962)MathSciNetCrossRefGoogle Scholar
  45. 45.
    P. Erdős, C.A. Rogers, The star number of coverings of space with convex bodies. Acta Arith. 9, 41–45 (1964)MathSciNetCrossRefGoogle Scholar
  46. 46.
    G. Fejes Tóth, A note on covering by convex bodies. Canad. Math. Bull. 52(3), 361–365 (2009)MathSciNetCrossRefGoogle Scholar
  47. 47.
    R.J. Gardner, Geometric tomography, in Encyclopedia of Mathematics and its Applications, vol. 58 (Cambridge University Press, 2006)Google Scholar
  48. 48.
    E.D. Gluskin, A.E. Litvak, A remark on vertex index of the convex bodies, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, vol. 2050, pp. 255–265Google Scholar
  49. 49.
    I.T. Gohberg, A.S. Markus, A certain problem about the covering of convex sets with homothetic ones, in Izvestiya Moldavskogo Filiala Akademii Nauk SSSR, vol. 10/76 (1960), pp. 87–90 (In Russian)Google Scholar
  50. 50.
    B. Grünbaum, Borsuk’s problem and related questions, in Proceedings of Symposia in Pure Mathematics, vol. 7 (American Mathematical Society, Providence, RI, 1963)Google Scholar
  51. 51.
    H. Hadwiger, Ungelöste Probleme Nr. 20. Elem. der Math.12, 121 (1957)Google Scholar
  52. 52.
    H. Hadwiger, Ungelöste Probleme Nr. 38. Elem. der Math. 15, 130–131 (1960)Google Scholar
  53. 53.
    H. He, H. Martini, S. Wu, On covering functionals of convex bodies. J. Math. Anal. Appl. 437(2), 1236–1256 (2016)MathSciNetCrossRefGoogle Scholar
  54. 54.
    J. Kahn, G. Kalai, A counterexample to Borsuk’s conjecture. Bull. Am. Math. Soc. 29, 60–62 (1993)MathSciNetCrossRefGoogle Scholar
  55. 55.
    G. Kiss, Illumination problems and codes. Period. Math. Hungar. 39, 65–71 (1999)Google Scholar
  56. 56.
    G. Kiss, P.O. de Wet, Notes on the illumination parameters of convex polytopes. Contrib. Discrete Math.7/1, 58–67 (2009)Google Scholar
  57. 57.
    M. Lassak, Solution of Hadwiger’s covering problem for centrally symmetric convex bodies in \({\mathbb{E}}^3\). J. Lond. Math. Soc. 30, 501–511 (1984)CrossRefGoogle Scholar
  58. 58.
    M. Lassak, Covering a plane convex body by four homothetical copies with the smallest positive ratio. Geom. Dedicata 21, 157–167 (1986)MathSciNetCrossRefGoogle Scholar
  59. 59.
    M. Lassak, Covering the boundary of a convex set by tiles. Proc. Am. Math. Soc. 104, 269–272 (1988)MathSciNetCrossRefGoogle Scholar
  60. 60.
    M. Lassak, Illumination of three-dimensional convex bodies of constant width, in Proceedings of the 4th International Congress of Geometry (Aristotle University of Thessaloniki, 1997), pp. 246–250Google Scholar
  61. 61.
    M. Lassak, H. Martini, M. Spirova, On translative coverings of convex bodies. Rocky Mountain J. Math. 44(4), 1281–1299 (2014)MathSciNetCrossRefGoogle Scholar
  62. 62.
    F.W. Levi, Überdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns. Arch. Math. 6(5), 369–370 (1955)MathSciNetCrossRefGoogle Scholar
  63. 63.
    H. Martini, Some results and problems around zonotopes, in Colloquia Mathematica Societatis Janos Bolyai, Intuitive Geometry, Sioófok 1985, vol. 48 (North Holland, 1987), pp. 383–418Google Scholar
  64. 64.
    H. Martini, V. Soltan, Combinatorial problems on the illumination of convex bodies. Aequationes Math. 57, 121–152 (1999)MathSciNetCrossRefGoogle Scholar
  65. 65.
    H. Martini, C. Richter, M. Spirova, Illuminating and covering convex bodies. Discrete Math. 337, 106–118 (2014)MathSciNetCrossRefGoogle Scholar
  66. 66.
    M. Naszódi, Fractional illumination of convex bodies. Contrib. Discrete Math. 4(2), 83–88 (2009)MathSciNetzbMATHGoogle Scholar
  67. 67.
    M. Naszódi, A spiky ball. Mathematika 62(2), 630–636 (2016)MathSciNetCrossRefGoogle Scholar
  68. 68.
    I. Papadoperakis, An estimate for the problem of illumination of the boundary of a convex body in \({\mathbb{E}}^3\). Geom. Dedicata 75, 275–285 (1999)MathSciNetCrossRefGoogle Scholar
  69. 69.
    C.A. Rogers, A note on coverings. Mathematika 4, 1–6 (1957)MathSciNetCrossRefGoogle Scholar
  70. 70.
    C.A. Rogers, G.C. Shephard, The difference body of a convex body. Arch. Math. 8, 220–233 (1957)MathSciNetCrossRefGoogle Scholar
  71. 71.
    C.A. Rogers, C. Zong, Covering convex bodies by translates of convex bodies. Mathematika 44, 215–218 (1997)MathSciNetCrossRefGoogle Scholar
  72. 72.
    R. Schneider, Convex Bodies: The Brun-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, vol. 151, 2nd edn. (Cambridge University Press, Cambridge, UK, 2014)Google Scholar
  73. 73.
    O. Schramm, Illuminating sets of constant width. Mathematika 35, 180–189 (1988)MathSciNetCrossRefGoogle Scholar
  74. 74.
    P. Soltan, Helly’s theorem for \(d\)-convex sets (in Russian). Dokl. Akad. Nauk. SSSR 205/3, 537–539 (1972)Google Scholar
  75. 75.
    P. Soltan, V. Soltan, Illumination through convex bodies. Dokl. Akad. Nauk. SSSR 286, 50–53 (1986)MathSciNetzbMATHGoogle Scholar
  76. 76.
    K.J. Swanepoel, Quantitative illumination of convex bodies and vertex degrees of geometric Steiner minimal trees. Mathematika 52(1-2), 47–52 (2005)MathSciNetCrossRefGoogle Scholar
  77. 77.
    I. Talata, Solution of Hadwiger-Levi’s covering problem for duals of cyclic \(2k\)-polytopes. Geom. Dedicata 74, 61–71 (1999)MathSciNetCrossRefGoogle Scholar
  78. 78.
    R.G. Trelford, Separation in and X-raying of convex bodies, Ph.D. thesis, University of Calgary, 2014, pp. 1–126Google Scholar
  79. 79.
    B. Weissbach, Invariante Beleuchtung konvexer Körper. Beiträge Alg. Geom. 37, 9–15 (1996)zbMATHGoogle Scholar
  80. 80.
    S. Wu, Two equivalent forms of Hadwiger’s covering conjecture. Sci. Sin. Math. 44(3), 275–285 (2014). (in Chinese)CrossRefGoogle Scholar
  81. 81.
    S. Wu, Upper bounds for the covering number of centrally symmetric convex bodies in \({\mathbb{R}}^{n}\). Math. Inequal. Appl. 17(4), 1281–1298 (2014)MathSciNetzbMATHGoogle Scholar
  82. 82.
    L. Yu, C. Zong, On the blocking number and the covering number of a convex body. Adv. Geom. 9(1), 13–29 (2009)MathSciNetCrossRefGoogle Scholar
  83. 83.
    L. Yu, Blocking numbers and fixing numbers of convex bodies. Discrete Math. 309, 6544–6554 (2009)MathSciNetCrossRefGoogle Scholar
  84. 84.
    C. Zong, Some remarks concerning kissing numbers, blocking numbers and covering numbers. Period. Math. Hungar. 30(3), 233–238 (1995)MathSciNetCrossRefGoogle Scholar
  85. 85.
    C. Zong, A quantitative program for Hadwigers covering conjecture. Sci. China Math. 53(9), 2551–2560 (2010)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of PannoniaVeszprémHungary

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