View More View Less
  • 1 Budapest University of Technology, Egry József utca 1, Budapest, 1111, Hungary
  • 1 Hungarian Academy of Sciences, Műegyetem rakpart 1-3., K.261, Budapest, 1111, Hungary
  • 2 Eötvös Lóránd University, Budapest, Hungary
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

Hujter and Lángi defined the k-fold Borsuk number of a set S in Euclidean n-space of diameter d > 0 as the smallest cardinality of a family F of subsets of S, of diameters strictly less than d, such that every point of S belongs to at least k members of F.

We investigate whether a k-fold Borsuk covering of a set S in a finite dimensional real normed space can be extended to a completion of S. Furthermore, we determine the k-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes.

  • [1]

    Boltyanski, V. G., On the partition of plane figures into pieces of smaller diameters (in Russian), Colloq. Math., 21 (1970), 253263.

    • Search Google Scholar
    • Export Citation
  • [2]

    Boltyanski V. G. and Gohberg, I. C., The Decomposition of Figures into Smaller Parts, translated from Russian, The University of Chicago Press, Chicago, 1980.

    • Search Google Scholar
    • Export Citation
  • [3]

    Boltyanski, V., Martini, H. and Soltan, P. S., Excursions into Combinatorial Geometry, Springer-Verlag, Berlin Heidelberg, 1997.

  • [4]

    Boltyanski, V. G. and Soltan, V., Borsuk’s problem (in Russian), Mat. Zametki 22 (1977), 621631.

  • [5]

    Borsuk, K., Drei Sätze über die n-dimensionale eukildische Sphäre, Fundamenta Math., 20 (1933), 177190.

  • [6]

    Eggleston, H. G., Covering a three-dimensional set with sets of smaller diameter, J. London Math. Soc., 30 (1955), 1124.

  • [7]

    Eggleston, H. G., Sets of constant width in finite dimensional Banach spaces, Israel J. Math., 3 (1965), 163172.

  • [8]

    Grünbaum, B., Borsuk’s partition conjecture in Minkowski planes, Bull. Res. Council Israel, F1, N1,(1957), 2530.

  • [9]

    Grünbaum, B., A simple proof of Borsuk’s conjecture in three dimensions, Proc. Cambridge Philos. Soc., 53 (1957), 776778.

  • [10]

    Hammer, P. C., Convex curves of constant Minkowski breadth, Proc. Symp. Pure Math., 7, Amer. Math. Soc., Providence RI (1963), 291304.

    • Search Google Scholar
    • Export Citation
  • [11]

    Heppes, A., On the partitioning of a three-dimensional point set into sets of smaller diameter (Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 7 (1957), 413416.

    • Search Google Scholar
    • Export Citation
  • [12]

    Hujter, M. and Lángi, Z., On the multiple Borsuk numbers of sets, Israel J. Math., 199 (2014), 219239.

  • [13]

    Joós, A. and Lángi, Z., On the relative distances of seven points in a plane convex body, J. Geom., 87 (2007), 8395.

  • [14]

    Kahn, J. and Kalai, G., A counterexample to Borsuk’s conjecture, Bull. Amer. Math. Soc., 29 (1993), 6062.

  • [15]

    Lassak, M., On relatively equilateral polygons inscribed in a convex body, Publ. Math. Debrecen 65 (2004), 133148.

  • [16]

    Moreno, J.P., Porosity and unique completion in strictly convex spaces, Math. Z. 267(1-2) (2011), 173-184.

  • [17]

    Naszódi, M. and Visy, B., Sets with a unique extension to a set of constant width, Discrete geometry, Monogr. Textbooks Pure Appl. Math., 253, Dekker, New York, 2003, 373380.

    • Search Google Scholar
    • Export Citation
  • [18]

    Raigorodskii, A.M., Around Borsuk’s hypothesis, Journal of Mathematical Sciences 154(2008), no. 4, 604623 (English).

  • [19]

    Sallee, G.T., The maximal set of constant width in a lattice, Pacific J. Math. 28 (1969), 669674.

  • [20]

    Stahl, S., n-tuple colorings and associated graphs, J. Combin. Theory Ser. B 20 (1976), 185203.

  • [21]

    Thompson, A.C., Minkowski Geometry, Encyclopedia of Mathematics and its Applications 63, Cambridge University Press, Cambridge, 1996.

  • [22]

    Yost, D., Irreducible convex sets, Mathematika, 38(1991), no. 1, 134155.

The author instruction is available in PDF.

Please, download the file from HERE

Manuscript submission: HERE


  • Impact Factor (2019): 0.486
  • Scimago Journal Rank (2019): 0.234
  • SJR Hirsch-Index (2019): 23
  • SJR Quartile Score (2019): Q3 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.309
  • Scimago Journal Rank (2018): 0.253
  • SJR Hirsch-Index (2018): 21
  • SJR Quartile Score (2018): Q3 Mathematics (miscellaneous)

Language: English, French, German

Founded in 1966
Publication: One volume of four issues annually
Publication Programme: 2020. Vol. 57.
Indexing and Abstracting Services:

  • CompuMath Citation Index
  • Mathematical Reviews
  • Referativnyi Zhurnal/li>
  • Research Alert
  • Science Citation Index Expanded (SciSearch)/li>
  • The ISI Alerting Services


Subscribers can access the electronic version of every printed article.

Senior editors

Editor(s)-in-Chief: Pálfy Péter Pál

Managing Editor(s): Sági, Gábor

Editorial Board

  • Biró, András (Number theory)
  • Csáki, Endre (Probability theory and stochastic processes, Statistics)
  • Domokos, Mátyás (Algebra (Ring theory, Invariant theory))
  • Győri, Ervin (Graph and hypergraph theory, Extremal combinatorics, Designs and configurations)
  • O. H. Katona, Gyula (Combinatorics)
  • Márki, László (Algebra (Semigroup theory, Category theory, Ring theory))
  • Némethi, András (Algebraic geometry, Analytic spaces, Analysis on manifolds)
  • Pach, János (Combinatorics, Discrete and computational geometry)
  • Rásonyi, Miklós (Probability theory and stochastic processes, Financial mathematics)
  • Révész, Szilárd Gy. (Analysis (Approximation theory, Potential theory, Harmonic analysis, Functional analysis))
  • Ruzsa, Imre Z. (Number theory)
  • Soukup, Lajos (General topology, Set theory, Model theory, Algebraic logic, Measure and integration)
  • Stipsicz, András (Low dimensional topology and knot theory, Manifolds and cell complexes, Differential topology)
  • Szász, Domokos (Dynamical systems and ergodic theory, Mechanics of particles and systems)
  • Tóth, Géza (Combinatorial geometry)

Gábor Sági
Address: P.O. Box 127, H–1364 Budapest, Hungary
Phone: (36 1) 483 8344 ---- Fax: (36 1) 483 8333