A note on Borsuk’s problem in Minkowski spaces

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In 1993, Kahn and Kalai famously constructed a sequence of finite sets in d-dimensional Euclidean spaces that cannot be partitioned into less than  parts of smaller diameter. Their method works not only for the Euclidean, but for all lp-spaces as well. In this short note, we observe that the larger the value of p, the stronger this construction becomes.

Sobre autores

A. Raigorodskii

Moscow Institute of Physics and Technology; Moscow State University; Caucasus Mathematical Center, Adyghe State University; Buryat State University

Autor responsável pela correspondência
Email: mraigor@yandex.ru
Rússia, Moscow; Moscow; Maykop; Ulan-Ude

A. Sagdeev

Alfred Renyi Institute of Mathematics; Moscow Institute of Physics and Technology

Email: sagdeevarsenii@gmail.com
Hungria, Budapest; Moscow, Russia

Bibliografia

  1. Baker R.C., Harman G., Pintz J. The difference between consecutive primes. II // Proc. Lond. Math. Soc. (3). 2001. Vol. 83. No. 3. P. 532–562.
  2. Bogolubsky L.I., Raigorodskii A.M. On bounds in Borsuk’s problem // Proc. MIPT (Trudy MFTI). 2019. Vol. 11. No. 3. P. 20–49.
  3. Boltyanskii V.G., Gohberg I.T. Results and Problems in Combinatorial Geometry. Cambridge: Cambridge Univ. Press, 1985; M.: Nauka, 1965.
  4. Bondarenko A. On Borsuk’s Conjecture for Two-Distance Sets // Discrete Comput. Geom. 2014. Vol. 51. No. 3. P. 509–515.
  5. Borsuk K. Drei Sätze über die -dimensionale euklidische Sphäre // Fundamenta Math. 1933. Vol. 20. P. 177–190.
  6. Bourgain J., Lindenstrauss J. On covering a set in by balls of the same diameter, Geometric Aspects of Functional Analysis (J. Lindenstrauss and V. Milman, eds.) // Lecture Notes in Math. 1469. Springer, Berlin, 1991. P. 138–144.
  7. Frankl P., Wilson R.M. Intersection theorems with geometric consequences // Combinatorica. 1981. Vol. 1. No. 4. P. 357–368.
  8. Grünbaum B. Borsuk’s partition conjecture in Minkowski planes // Bull. Res. Council Israel Sect. F. 7F. 1957. P. 25–30.
  9. Jenrich T., Brouwer A.E. A 64-Dimensional Counterexample to Borsuk’s Conjecture // Electron. J. Combin. 2014. Vol. 21. No. 4. P4.29.
  10. Kahn J., Kalai G. A counterexample to Borsuk’s conjecture // Bull. Amer. Math. Soc. 1993. Vol. 29. No. 1. P. 60–62.
  11. Lassak M. An estimate concerning Borsuk’s partition problem // Bull. Acad. Polon. Sci. Ser. Math. 1982. Vol. 30. P. 449–451.
  12. Raigorodskii A.M. Semidefinite programming in combinatorial optimization with applications to coding theory and geometry. 2013. Thesis. Available at https://theses.hal.science/tel-00948055
  13. Raigorodskii A.M. Around Borsuk’s conjecture // J. Math. Sci. 2008. Vol. 154. No. 4. P. 604–623.
  14. Raigorodskii A.M. Coloring Distance Graphs and Graphs of Diameters // Thirty Essays on Geometric Graph Theory / ed. J. Pach. Springer, 2013. P. 429–460.
  15. Raigorodskii A.M. On a bound in Borsuk’s problem // Russian Math. Surveys, 1999. Vol. 54. No. 2. P. 453–454.
  16. Raigorodskii A.M. On the dimension in Borsuk’s problem // Russian Math. Surveys. 1997. Vol. 52. No. 6. P. 1324–1325.
  17. Rogers C.A., C. Zong C. Covering convex bodies by translates of convex bodies // Mathematika. 1997. Vol. 44. No. 1. P. 215–218.
  18. Schramm O. Illuminating sets of constant width // Mathematika. 18. Vol. 35. P. 180–189.
  19. Swanepoel K.J. Combinatorial distance geometry in normed spaces // New Trends in Intuitive Geometry. 2018. P. 407–458.
  20. Wang J., Xue F. Borsuk’s partition problem in four-dimensional space. Preprint arXiv:2206.15277.
  21. Weissbach B. Sets with large Borsuk number // Beiträge Alg. Geom. 2000. Vol. 41. P. 417–423.
  22. Yu L., Zong C. On the blocking number and the covering number of a convex body // Adv. Geom. 2009. Vol. 9. No. 1. P. 13–29.

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