- PII
- 10.31857/S2686954324010151-1
- DOI
- 10.31857/S2686954324010151
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume 515 / Issue number 1
- Pages
- 100-104
- Abstract
- 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.
- Keywords
- Date of publication
- 15.11.2024
- Year of publication
- 2024
- Number of purchasers
- 0
- Views
- 45
References
- 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.
