Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On some covering problems in geometry

Author: Márton Naszódi
Journal: Proc. Amer. Math. Soc. 144 (2016), 3555-3562
MSC (2010): Primary 52C17, 05B40, 52A23
Published electronically: January 26, 2016
MathSciNet review: 3503722
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $ n$-space with translates of a convex body, or more generally, any measurable set. We obtain a bound for the density of covering the $ n$-sphere by rotated copies of a spherically convex set (or, any measurable set). Using the same method, we sharpen an estimate by Artstein-Avidan and Slomka on covering a bounded set by translates of another.

The main novelty of our method is that it is not probabilistic. The key idea, which makes our proofs rather simple and uniform through different settings, is an algorithmic result of Lovász and Stein.

References [Enhancements On Off] (What's this?)

  • [AAR11] S. Artstein-Avidan and O. Raz, Weighted covering numbers of convex sets, Adv. Math. 227 (2011), no. 1, 730-744. MR 2782207 (2012g:52003),
  • [AAS13] Shiri Artstein-Avidan and Boaz A. Slomka, On weighted covering numbers and the levi-hadwiger conjecture, arXiv:1310.7892 [math] (2013-10).
  • [BW03] Károly Böröczky Jr. and Gergely Wintsche, Covering the sphere by equal spherical balls, Discrete and computational geometry, Algorithms Combin., vol. 25, Springer, Berlin, 2003, pp. 235-251. MR 2038476 (2005e:52028),
  • [Dum07] Ilya Dumer, Covering spheres with spheres, Discrete Comput. Geom. 38 (2007), no. 4, 665-679. MR 2365829 (2008m:52042),
  • [ER61] P. Erdős and C. A. Rogers, Covering space with convex bodies, Acta Arith. 7 (1961/1962), 281-285. MR 0149373 (26 #6863)
  • [FK08] Z. Füredi and J.-H. Kang, Covering the $ n$-space by convex bodies and its chromatic number, Discrete Math. 308 (2008), no. 19, 4495-4500. MR 2433777 (2009c:52031),
  • [FT09] Gábor Fejes Tóth, A note on covering by convex bodies, Canad. Math. Bull. 52 (2009), no. 3, 361-365.
  • [Für88] Zoltán Füredi, Matchings and covers in hypergraphs, Graphs Combin. 4 (1988), no. 2, 115-206.
  • [Lov75] L. Lovász, On the ratio of optimal integral and fractional covers, Discrete Math. 13 (1975), no. 4, 383-390.
  • [Mat02] Jiří Matoušek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002.
  • [MP00] V. D. Milman and A. Pajor, Entropy and asymptotic geometry of non-symmetric convex bodies, Adv. Math. 152 (2000), no. 2, 314-335. MR 1764107 (2001e:52004),
  • [Nas09] Márton Naszódi, Fractional illumination of convex bodies, Contrib. Discrete Math. 4 (2009), no. 2, 83-88. MR 2592425 (2011d:52038)
  • [Rog57] C. A. Rogers, A note on coverings, Mathematika 4 (1957), 1-6. MR 0090824 (19,877c)
  • [Rog63] C. A. Rogers, Covering a sphere with spheres, Mathematika 10 (1963), 157-164. MR 0166687 (29 #3960)
  • [Rog64] C. A. Rogers, Packing and covering, Cambridge Tracts in Mathematics and Mathematical Physics, No. 54, Cambridge University Press, New York, 1964. MR 0172183 (30 #2405)
  • [Sch88] Oded Schramm, Illuminating sets of constant width, Mathematika 35 (1988), no. 2, 180-189. MR 986627 (89m:52013),
  • [Ste56] S. Stein, The symmetry function in a convex body, Pacific J. Math. 6 (1956), 145-148. MR 0080321 (18,228g)
  • [Ste74] S. K. Stein, Two combinatorial covering theorems, J. Combinatorial Theory Ser. A 16 (1974), 391-397. MR 0340062 (49 #4818)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 52C17, 05B40, 52A23

Retrieve articles in all journals with MSC (2010): 52C17, 05B40, 52A23

Additional Information

Márton Naszódi
Affiliation: Department of Geometry, Lorand Eötvös University, Pázmány Péter Sétány 1/C Budapest, Hungary 1117

Keywords: Covering, Rogers' bound, spherical cap, density, set-cover
Received by editor(s): March 30, 2015
Received by editor(s) in revised form: October 1, 2015
Published electronically: January 26, 2016
Additional Notes: The author acknowledges the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and the Hung. Nat. Sci. Found. (OTKA) grant PD104744.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society