Abstract
In 1957, Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2n translates of its interior. Up to now, this conjecture is still open for all n ⩾ 3. In 1933, Borsuk made a conjecture that every n-dimensional bounded set can be divided into n + 1 subsets of smaller diameters. Up to now, this conjecture is open for 4 ⩽ n ⩽ 297. In this article we encode the two conjectures into continuous functions defined on the spaces of convex bodies, propose a four-step program to attack them, and obtain some partial results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Belousov J F. Theorems on the covering of plane figures. Ukrain Geom Sb, 1977, 20: 10–17
Besicovitch A S. Measure of asymmetry of convex curves. J London Math Soc, 1948, 23: 237–240
Bezdek K. The illumination conjecture and its extensions. Period Math Hungar, 2006, 53: 59–69
Bezdek K, Bisztriczky T. A proof of Hadwiger’s covering conjecture for dual cyclic polytopes. Geom Dedicata, 1997, 68: 29–41
Boltyanski V G. Solution of the illumination problem for bodies with md M = 2. Discrete Comput Geom, 2001, 26: 527–541
Boltyanski V G, Martini H. Illumination of direct vector sums of convex bodies. Studia Sci Math Hungar, 2007, 44: 367–376
Boltyanski V G, Martini H, Soltan P S. Excursions into Combinatorial Geometry. Berlin: Springer-Verlag, 1997
Böröczky Jr K, Wintsche G. Covering the sphere by equal spherical balls. In: The Goodman-Pollack Festschrift. Berlin: Springer-Verlag, 2003, 237–253
Borsuk K. Drei Sätzeüber die n-dimensionale Euklidische sphäre. Fund Math, 1933, 20: 177–190
Brass P, Moser W, Pach J. Research Problems in Discrete Geometry. New York: Springer-Verlag, 2005
Chakerian G D, Groemer H. Convex bodies of constant width. In: Gruber P M, Wills J M, eds. Convexity and its Applications. Basel: Birkhäuser, 1983, 49–96
Eggleston H G. Convexity. Cambridge: Cambridge University Press, 1958
Fejes Tóth L. Kreisüberdeckungen der hyperbolischen Ebene. Acta Math Acad Sci Hungar, 1953, 4: 111–114
Gruber P M. Convex and Discrete Geometry. Berlin: Springer-Verlag, 2007
Grünbaum B. Borsuk’s problem and related questions. Proc Symp Pure Math, 1963, 7: 271–284
Hadwiger H. Ungelöste Probleme No. 20. Elem Math, 1957, 12: 121
Hinrichs A, Richter C. New sets with large Borsuk numbers. Discrete Math, 2003, 270: 137–147
John F. Extremum problems with inequalities as subsidiary conditions. In: Courant Anniversary Volume. New York: Interscience, 1948, 187–204
Kahn J, Kalai G. A counterexample to Borsuk’s conjecture. Bull Amer Math Soc, 1993, 29: 60–62
Krotoszynski S. Covering a plane convex body with five smaller homothetical copies. Beiträge Algebra Geom, 1987, 25: 171–176
Lassak M. Solution of Hadwiger’s covering problem for centrally symmetric convex bodies in E 3. J London Math Soc, 1984, 30: 501–511
Lassak M. Covering a plane convex body by four homothetical copies with the smallest positive ratio. Geom Dedicata, 1986, 21: 157–167
Levi F W. Ein geometrisches Überdeckungsproblem. Arch Math, 1954, 5: 476–478
Martini H, Soltan V. Combinatorial problems on the illumination of convex bodies. Aequationes Math, 1999, 57: 121–152
Papadoperakis I. An estimate for the problem of illumination of the boundary of a convex body in E 3. Geom Dedicata, 1999, 75: 275–285
Perkal J. Sur la subdivision des ensembles en parties de diamétre inferieur. Colloq Math, 1947, 1: 45
Rogers C A, Zong C. Covering convex bodies by translates of convex bodies. Mathematika, 1997, 44: 215–218
Schramm O. Illuminating sets of constant width. Mathematika, 1988, 35: 180–189
Schütte K. Überdeckungen der Kugel mit höchstens acht Kreisen. Math Ann, 1955, 129: 181–186
Talata I. Solution of Hadwiger-Levi’s covering problem for duals of cyclic 2k-polytopes. Geom Dedicata, 1999, 74: 61–71
Zong C. Some remarks concerning kissing numbers, blocking numbers and covering numbers. Period Math Hungar, 1995, 30: 233–238
Zong C. The kissing number, blocking number and covering number of a convex body. Contemp Math, 2008, 453: 529–548
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Wang Yuan on the Occasion of his 80th Birthday
Rights and permissions
About this article
Cite this article
Zong, C. A quantitative program for Hadwiger’s covering conjecture. Sci. China Math. 53, 2551–2560 (2010). https://doi.org/10.1007/s11425-010-4087-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-4087-3