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Flavors of Translative Coverings

  • Márton NaszódiEmail author
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)

Abstract

We survey results on the problem of covering the space \(\mathbb R^n\), or a convex body in it, by translates of a convex body. Our main goal is to present a diverse set of methods. A theorem of Rogers is a central result, according to which, for any convex body K, the space \(\mathbb R^n\) can be covered by translates of K with density around \(n\ln n\). We outline four approaches to proving this result. Then, we discuss the illumination conjecture, decomposability of multiple coverings, Sudakov’s inequality and some problems concerning coverings by sequences of sets.

Keywords

Covering Rogers’ bound Spherical cap Density Set-cover Illumination Borsuk’s conjecture Multiple covering Sudakov’s inequality 

2010 Mathematics Subject Classification

52C17 05B40 52A23 

Notes

Acknowledgements

The author is grateful for the many illuminating conversations with Gábor Fejes Tóth about covering problems in general, and about this manuscript.

References

  1. 1.
    N. Alon, J.H. Spencer, The Probabilistic Method. With an Appendix on the Life and Work of Paul Erdős, 3rd ed. (Hoboken, Wiley, 2008) (English)Google Scholar
  2. 2.
    S. Artstein, V. Milman, S.J. Szarek, Duality of metric entropy. Ann. Math. (2) 159(3), 1313–1328 (2004). MR2113023 (2005h:47037)MathSciNetzbMATHGoogle Scholar
  3. 3.
    S. Artstein-Avidan, A. Giannopoulos, Mathematical surveys and monographs, Asymptotic Geometric Analysis. Part I (American Mathematical Society, Providence, 2015). MR3331351Google Scholar
  4. 4.
    S. Artstein-Avidan, O. Raz, Weighted covering numbers of convex sets. Adv. Math. 227(1), 730–744 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    S. Artstein-Avidan, B.A. Slomka, On Weighted Covering Numbers and the Levi-Hadwiger Conjecture. Israel Journal of Mathematics 209(1), 125–155 (September 2015), arXiv:1310.7892MathSciNetzbMATHGoogle Scholar
  6. 6.
    K. Ball, Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. (2) 44(2), 351–359 (1991). MR1136445 (92j:52013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    K. Ball, Volumes of sections of cubes and related problems, Geometric Aspects of Functional Analysis (1987–88) (1989), pp. 251–260. MR1008726 (90i:52019)Google Scholar
  8. 8.
    K. Bezdek, Classical Topics in Discrete Geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (Springer, New York, 2010)zbMATHGoogle Scholar
  9. 9.
    K. Bezdek, Gy 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). MR2547800 (2010m:52009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    K. Bezdek, The illumination conjecture and its extensions. Period. Math. Hung. 53(1–2), 59–69 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    K. Bezdek, The problem of illumination of the boundary of a convex body by affine subspaces. Mathematika 38(2), 362–375 (1991, 1992). MR1147835 (92m:52020)MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. Bezdek, K. Bezdek, Eine hinreichende Bedingung für die Überdeckung des Einheitswürfels durch homothetische Exemplare im n- dimensionalen euklidischen Raum, Beiträge Algebra Geom. 17 (1984), pp. 5–21. MR755762 (85h:52017)Google Scholar
  13. 13.
    K. Bezdek, M.A. Khan, The geometry of homothetic covering and illumination, in Discrete geometry and symmetry (to appear), 2018Google Scholar
  14. 14.
    K. Bezdek, Z. Lángi, M. Naszódi, P. Papez, Ball-polyhedra. Discret. Comput. Geom. 38(2), 201–230 (2007). MR2343304 (2008i:52001)MathSciNetzbMATHGoogle Scholar
  15. 15.
    V. Boltyanski, H. Martini, P.S. Soltan, Excursions into Combinatorial Geometry (Universitext, Springer, Berlin, 1997). MR1439963 (98b:52001)zbMATHGoogle Scholar
  16. 16.
    V. Boltyanski, The problem of illuminating the boundary of a convex body. Izv. Mold. Fil. AN SSSR 76, 77–84 (1960)Google Scholar
  17. 17.
    T. Bonnesen, W. Fenchel, Theory of convex bodies. Transl. from the German and ed. by L. Boron, C. Christenson, B. Smith, with the Collaboration of W. Fenchel (Moscow, Idaho, USA): BCS Associates. IX, 172 p. (1987) (English)Google Scholar
  18. 18.
    K. Böröczky Jr., Finite Packing and Covering, Cambridge Tracts in Mathematics, vol 154 (Cambridge University Press, Cambridge, 2004). MR2078625 (2005g:52045)Google Scholar
  19. 19.
    K. Böröczky Jr., K. Böröczky Jr., Covering the spere by equal spherical balls, Discrete and Computational Geometry (Springer, Berlin, 2003), pp. 235–251Google Scholar
  20. 20.
    K. Borsuk, Drei sätze über die n-dimensionale euklidische sphäre, Fundamenta Mathematicae 20 (1933), no. 1, 177–190 (ger)zbMATHGoogle Scholar
  21. 21.
    P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry (Springer, New York, 2005)Google Scholar
  22. 22.
    H.S.M. Coxeter, L. Few, C.A. Rogers, Covering space with equal spheres. Mathematika 6, 147–157 (1959). MR0124821 (23 #A2131)MathSciNetzbMATHGoogle Scholar
  23. 23.
    B.V. Dekster, Each convex body in \(E^{3}\) symmetric about a plane can be illuminated by 8 directions. J. Geom. 69(1–2), 37–50 (2000). MR1800455 (2001m:52003)Google Scholar
  24. 24.
    I. Dumer, Covering spheres with spheres, Discrete Comput. Geom. 38(4), 665–679 (2007)MathSciNetzbMATHGoogle Scholar
  25. 25.
    I. Dumer, Covering spheres with spheres (2018), arXiv:0606002v2 [math]
  26. 26.
    H.G. Eggleston, Covering a three-dimensional set with sets of smaller diameter. J. Lond. Math. Soc. 30, 11–24 (1955). MR0067473 (16,734b)MathSciNetzbMATHGoogle Scholar
  27. 27.
    P. Erdős and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, Infinite and finite sets (Colloquium, Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. II, 1975, pp. 609–627. Colloquium Mathematical Society, János Bolyai, vol. 10. MR0382050 (52 #2938)Google Scholar
  28. 28.
    P. Erdős, C.A. Rogers, Covering space with convex bodies. Acta Arith. 7, 281–285 (1961/1962)MathSciNetzbMATHGoogle Scholar
  29. 29.
    G. Fejes Tóth, A note on covering by convex bodies. Can. Math. Bull. 52(3), 361–365 (2009)Google Scholar
  30. 30.
    L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, (Band LXV, Springer, Berlin, 1953). MR0057566 (15,248b)Google Scholar
  31. 31.
    G. Fejes Tóth, New results in the theory of packing and covering, in Convexity and its Applications, Collect. Surv. (1983), 318–359; 1983 (English)Google Scholar
  32. 32.
    G. Fejes Tóth, Packing and covering, Handbook of Discrete and Computational Geometry, 2nd ed. (2004), pp. 25–53 (English)Google Scholar
  33. 33.
    L. Fejes Tóth, Personal communication (1984)Google Scholar
  34. 34.
    G. Fejes Tóth, Recent progress on packing and covering, in Advances in Discrete and Computational Geometry: Proceedings of the 1996 AMSIMS-SIAM Joint Summer Research Conference on Discrete and Computational Geometry: Ten Years Later, South Hadley, USA, 14–18 July 1996 (1996), pp. 145–162 (English)Google Scholar
  35. 35.
    G. Fejes Tóth, W. Kuperberg, A survey of recent results in the theory of packing and covering, New Trends in Discrete and Computational Geometry (1993), pp. 251–279. (English)Google Scholar
  36. 36.
    G. Fejes Tóth, W. Kuperberg, Packing and covering with convex sets, in Handbook of Convex Geometry, vol. B (1993), pp. 799–860 (English)Google Scholar
  37. 37.
    P. Frankl, R.M. Wilson, Intersection theorems with geometric consequences. Combinatorica 1(4), 357–368 (1981). MR647986 (84g:05085)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Z. Füredi, Matchings and covers in hypergraphs. Graphs Comb. 4(2), 115–206 (1988)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Z. Füredi, J.-H. Kang, Covering the n-space by convex bodies and its chromatic number. Discret. Math. 308(19), 4495–4500 (2008)Google Scholar
  40. 40.
    I. Gohberg, A. Markus, A problem on covering of convex figures by similar figures. Izv. Mold. Fil. AN SSSR 10, 87–90 (1960)Google Scholar
  41. 41.
    P. Gritzmann, Lattice covering of space with symmetric convex bodies. Mathematika 32(2), 311–315 (1985); (1986). MR834499MathSciNetzbMATHGoogle Scholar
  42. 42.
    H. Groemer, Covering and packing properties of bounded sequences of convex sets. Mathematika 29, 18–31 (1982). (English)MathSciNetzbMATHGoogle Scholar
  43. 43.
    H. Groemer, Coverings and packings by sequences of convex sets, Discrete Geometry and Convexity (New York, 1982); (1985), pp. 262–278Google Scholar
  44. 44.
    H. Groemer, Space coverings by translates of convex sets. Pac. J. Math. 82, 379–386 (1979). (English)MathSciNetzbMATHGoogle Scholar
  45. 45.
    B. Grünbaum, A simple proof of Borsuk’s conjecture in three dimensions. Math. Proc. Camb. Philos. Soc. 53, 776–778 (1957). MR0090072 (19,763d)MathSciNetzbMATHGoogle Scholar
  46. 46.
    H. Hadwiger, Ungelöste probleme, nr. 38. Elem. Math. 15, 130–131 (1960)Google Scholar
  47. 47.
    A. Heppes, On the partitioning of three-dimensional point-sets into sets of smaller diameter. Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 7, 413–416 (1957). MR0095450 (20 #1952)Google Scholar
  48. 48.
    A. Heppes, P. Révész, A splitting problem of Borsuk, Mat. Lapok 7 (1956), 108–111. MR0098353 (20 #4814)Google Scholar
  49. 49.
    M. Hujter, Z. Lángi, On the multiple Borsuk numbers of sets. Israel J. Math. 199(1), 219–239 (2014). MR3219534MathSciNetzbMATHGoogle Scholar
  50. 50.
    J. Januszewski, Translative covering a convex body by its homothetic copies. Stud. Sci. Math. Hung. 40(3), 341–348 (2003). MR2036964 (2005b:52044)MathSciNetzbMATHGoogle Scholar
  51. 51.
    E.jr Makai, J. Pach, Controlling function classes and covering Euclidean space. Stud. Sci. Math. Hung. 18, 435–459 (1983). (English)Google Scholar
  52. 52.
    J. Kahn, G. Kalai, A counterexample to Borsuk’s conjecture. Bull. Amer. Math. Soc. (N.S.) 29(1), 60–62 (1993). MR1193538 (94a:52007)MathSciNetzbMATHGoogle Scholar
  53. 53.
    M. Lassak, Illumination of three-dimensional convex bodies of constant width, in Proceedings of the 4th International Congress of Geometry: Thessaloniki, 1996 (1997), pp. 246–250. MR1470984 (98g:52013)Google Scholar
  54. 54.
    M. Lassak, Solution of Hadwiger’s covering problem for centrally symmetric convex bodies in \(E^{3}\), J. Lond. Math. Soc. (2) 30(3), 501–511 (1984). MR810959 (87e:52024)Google Scholar
  55. 55.
    M. Ledoux, M. Talagrand, Probability in Banach Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23 (Springer, Berlin, 1991). Isoperimetry and Processes. MR1102015 (93c:60001)zbMATHGoogle Scholar
  56. 56.
    F.W. Levi, Überdeckung eines Eibereiches durch Parallelverschiebung seines offenen Kerns. Arch. Math. (Basel) 6, 369–370 (1955). MR0076368 (17,888b)MathSciNetzbMATHGoogle Scholar
  57. 57.
    L. Lovász, On the ratio of optimal integral and fractional covers. Discret. Math. 13(4), 383–390 (1975)MathSciNetzbMATHGoogle Scholar
  58. 58.
    P. Mani-Levitska, J. Pach, Decomposition Problems for Multiple Coverings with Unit Balls (1986). Manuscript: Parts of the manuscript available at http://www.math.nyu.edu/~pach/publications/ unsplittable.pdf
  59. 59.
    H. Martini, V. Soltan, Combinatorial problems on the illumination of convex bodies. Aequ. Math. 57(2–3), 121–152 (1999)MathSciNetzbMATHGoogle Scholar
  60. 60.
    J. Math. Sci. Around Borsuk’s hypothesis. 154(4), 604–623 (2008). (English)Google Scholar
  61. 61.
    J. Matoušek, Lectures on discrete geometry (Graduate Texts in Mathematics), vol. 212 (Springer, New York, 2002)Google Scholar
  62. 62.
    A. Meir, L. Moser, On packing of squares and cubes. J. Comb. Theory 5, 126–134 (1968). MR0229142 (37 #4716)MathSciNetzbMATHGoogle Scholar
  63. 63.
    H. Minkowski, Allgemeine Lehrsätze über die convexen Polyeder. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1897, 198–219 (1897) (German)Google Scholar
  64. 64.
    M. Naszódi, Covering a set with homothets of a convex body. Positivity 14(1), 69–74 (2010). MR2596464MathSciNetzbMATHGoogle Scholar
  65. 65.
    M. Naszódi, Fractional illumination of convex bodies. Contrib. Discret. Math. 4(2), 83–88 (2009)Google Scholar
  66. 66.
    J. Pach, Covering the plane with convex polygons. Discret. Comput. Geom. 1, 73–81 (1986). (English)MathSciNetzbMATHGoogle Scholar
  67. 67.
    J. Pach, Decomposition of multiple packing and covering. Diskrete Geometrie, 2. Kolloq., Inst. Math. Univ. Salzburg 1980, 169–178 (1980). 1980 (English)Google Scholar
  68. 68.
    J. Pach, P.K. Agarwal, Combinatorial Geometry (New York, Wiley, 1995) (English)zbMATHGoogle Scholar
  69. 69.
    J. Pach, D. Pálvölgyi, Unsplittable coverings in the plane. Adv. Math. 302, 433–457 (2016)MathSciNetzbMATHGoogle Scholar
  70. 70.
    J. Pach, D. Pálvölgyi, G. Tóth, Survey on decomposition of multiple coverings, in Geometry—Intuitive, Discrete, and Convex (2013), pp. 219–257. MR3204561Google Scholar
  71. 71.
    J. Pach, G. Tóth, Decomposition of multiple coverings into many parts. Comput. Geom. 42(2), 127–133 (2009). (English)MathSciNetzbMATHGoogle Scholar
  72. 72.
    A. Pajor, N. Tomczak-Jaegermann, Remarques sur les nombres d’entropie d’un opérateur et de son transposé, C. R. Acad. Sci. Paris Sér. I Math. 301(15), 743–746 (1985). MR817602 (87f:47027)Google Scholar
  73. 73.
    D. Pálvölgyi, Indecomposable coverings with concave polygons. Discret. Comput. Geom. 44(3), 577–588 (2010). (English)MathSciNetzbMATHGoogle Scholar
  74. 74.
    D. Pálvölgyi, G. Tóth, Convex polygons are cover-decomposable. Discret. Comput. Geom. 43(3), 483–496 (2010). (English)MathSciNetzbMATHGoogle Scholar
  75. 75.
    I. Papadoperakis, An estimate for the problem of illumination of the boundary of a convex body in \(E^{3}\), Geom. Dedicata 75(3), 275–285 (1999). MR1689273 (2000g:52014)Google Scholar
  76. 76.
    J. Perkal, Sur la subdivision des ensembles en parties de diamétre intérieure. Colloq. Math. 1, 45 (1947)Google Scholar
  77. 77.
    A. Pietsch, Theorie der Operatorenideale (Zusammenfassung), Friedrich-Schiller-Universität, Jena, 1972. Wissenschaftliche Beiträge der Friedrich-Schiller-Universität Jena. MR0361822 (50 #14267)Google Scholar
  78. 78.
    Proc. Amer. Math. Soc. On some covering problems in geometry. 144(8), 3555–3562 (2016). MR3503722Google Scholar
  79. 79.
    J. Radon, Über eine Erweiterung des Begriffes der konvexen Funktionen mit einer Anwendung auf die Theorie der konvexen Körper. Wien. Ber. 125, 241–258 (1916). (German)Google Scholar
  80. 80.
    C.A. Rogers, A note on coverings. Mathematika 4, 1–6 (1957)MathSciNetzbMATHGoogle Scholar
  81. 81.
    C.A. Rogers, Covering a sphere with spheres. Mathematika 10, 157–164 (1963)MathSciNetzbMATHGoogle Scholar
  82. 82.
    C.A. Rogers, Lattice coverings of space. Mathematika 6, 33–39 (1959)MathSciNetzbMATHGoogle Scholar
  83. 83.
    C.A. Rogers, Packing and Covering, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 54 (Cambridge University Press, New York, 1964)Google Scholar
  84. 84.
    C.A. Rogers, G.C. Shephard, The difference body of a convex body. Arch. Math. (Basel) 8, 220–233 (1957)MathSciNetzbMATHGoogle Scholar
  85. 85.
    C.A. Rogers, C. Zong, Covering convex bodies by translates of convex bodies. Mathematika 44(1), 215–218 (1997)MathSciNetzbMATHGoogle Scholar
  86. 86.
    W. Schmidt, Maßtheorie in der Geometrie der Zahlen. Acta Math. 102, 159–224 (1959). (German)MathSciNetzbMATHGoogle Scholar
  87. 87.
    O. Schramm, Illuminating sets of constant width. Mathematika 35(2), 180–189 (1988)MathSciNetzbMATHGoogle Scholar
  88. 88.
    V. Soltan, É. Vásárhelyi, Covering a convex body by smaller homothetic copies. Geom. Dedicata 45(1), 101–113 (1993). MR1199732 (94a:52040)MathSciNetzbMATHGoogle Scholar
  89. 89.
    V. Soltan, Personal Communication (1990)Google Scholar
  90. 90.
    P.S. Soltan, V.P. Soltan, Illumination through convex bodies. Dokl. Akad. Nauk SSSR 286(1), 50–53 (1986). MR822098 (87f:52008)Google Scholar
  91. 91.
    J. Spencer, Asymptotic lower bounds for Ramsey functions. Discret. Math. 20(1), 69–76 (1977/78). MR0491337 (58 #10600)MathSciNetzbMATHGoogle Scholar
  92. 92.
    S.K. Stein, Two combinatorial covering theorems. J. Comb. Theory Ser. A 16(3), 391–397 (1974)MathSciNetzbMATHGoogle Scholar
  93. 93.
    V.N. Sudakov, Gaussian random processes, and measures of solid angles in Hilbert space, Dokl. Akad. Nauk SSSR 197, 43–45 (1971). MR0288832 (44 #6027)Google Scholar
  94. 94.
    W. Süss, Über den Vektorenbereich eines Eikörpers. Jahresber. Dtsch. Math.-Ver. 37, 87–90 (1928). (German)Google Scholar
  95. 95.
    L. Szabó, Recent results on illumination problems, in Intuitive Geometry (Budapest, 1995) (1997), pp. 207–221. MR1470759 (98h:52015)Google Scholar
  96. 96.
    M. Talagrand, A new isoperimetric inequality for product measure and the tails of sums of independent random variables. Geom. Funct. Anal. 1(2), 211–223 (1991). MR1097260 (92j:60004)MathSciNetzbMATHGoogle Scholar
  97. 97.
    G. Tardos, G. Toth, Multiple coverings of the plane with triangles. Discret. Comput. Geom. 38(2), 443–450 (2007). (English)MathSciNetzbMATHGoogle Scholar
  98. 98.
    Th. Estermann, Über den Vektorenbereich eines konvexen Körpers. Math. Z. 28, 471–475 (1928). (German)Google Scholar
  99. 99.
    N. Tomczak-Jaegermann, Dualité des nombres d’entropie pour des opérateurs à valeurs dans un espace de Hilbert. C. R. Acad. Sci. Paris Sér. I Math. 305(7), 299–301 (1987). MR910364 (89c:47027)Google Scholar
  100. 100.
    J.-L. Verger-Gaugry, Covering a ball with smaller equal balls in \({\mathbb{R}}^{n}\). Discret. Comput. Geom. 33(1), 143–155 (2005). (English)Google Scholar
  101. 101.
    B. Weissbach, Invariante Beleuchtung konvexer Körper, Beiträge Algebr. Geom. 37(1), 9–15 (1996). MR1407801 (97j:52011)Google Scholar
  102. 102.
    C.M. Zong, Some remarks concerning kissing numbers, blocking numbers and covering numbers. Period. Math. Hung. 30(3), 233–238 (1995). MR1334968MathSciNetzbMATHGoogle Scholar
  103. 103.
    C. Zong, The kissing number, blocking number and covering number of a convex body. Surv. Discret. Comput. Geom. 453, 529–548 (2008). MR2405694Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of GeometryLorand Eötvös UniversityBudapestHungary

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