Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Alimov A. R. Mazur Spaces and 4.3-intersection Property of (BM)-spaces. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 2, pp. 133-137. DOI: 10.18500/1816-9791-2016-16-2-133-137, EDN: WCNQGP

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
14.06.2016
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Language: 
Russian
Heading: 
Mathematics
UDC: 
517.982.252+517.982.256
DOI: 
10.18500/1816-9791-2016-16-2-133-137
EDN: 
WCNQGP

Mazur Spaces and 4.3-intersection Property of (BM)-spaces

Autors: 
Alimov A. R., Lomonosov Moscow State University
Abstract: 

The paper puts forward some combinatorial and geometric properties of finite-dimensional (BM)-spaces. A remarkable property of such spaces is that in these spaces one succeeds in giving an answer to some long-standing problems of geometric approximation theory, and in particular, to the question on the existence of continuous ε-selections on suns (Kolmogorov sets) for all ε > 0. A finite-dimensional polyhedral (BM)-space is shown to be a Mazur space, satisfies the 4.3-intersection property, and its unit ball is proved to be a generating set (in the sense of Polovinkin, Balashov, and Ivanov).

Key words: 
(BM)-space
4.3-intersection property
Mazur space
Mazur set
zonotope
generating set
References: 
  1. Brown A. L. Suns in normed linear spaces which are finite-dimensional // Math. Ann. 1987. Vol. 279. P. 87–101. DOI: https://doi.org/10.1007/BF01456192.
  2. Alimov A. R., Tsar’kov I. G. Connectedness and other geometric properties of suns and Chebyshev sets. Fund. prikl. matem., 2014, vol. 19, no. 4, pp. 21–91 (in Russian).
  3. Brown A. L. Suns in polyhedral spaces // Seminar of Math. Analysis. Proceedings / eds. D. G. Aґ lvarez, G. Lopez Acedo, R. V. Caro; Univ. Malaga and Seville (Spain), Sept. 2002 – Febr. 2003. Sevilla : Universidad de Sevilla, 2003. P. 139–146.
  4. Bednov B. B., Borodin P. A. Banach spaces that realize minimal fillings. Sb. Math., 2014, vol. 205, no. 4, pp. 3–20. DOI: https://doi.org/10.1070/SM2014v205n04ABEH004383
  5. Hansen A. B., Lima A. The structure of finite dimensional Banach spaces with the 3.2. intersection property // Acta Math. 1981. Vol. 146. P. 1–23. DOI: https://doi.org/10.1007/BF02392457.
  6. Boltyanskii V. G., Soltan P. S. Combinatorial geometry and convexity classes. Russian Math. Surveys, 1978, vol. 33, no. 1 (199), pp. 1–45. DOI: https://doi.org/10.1070/RM1978v033n01ABEH003730.
  7. Boltyanski V., Martini H., Soltan P. S. Excursions into combinatorial geometry. Berlin : Springer, 1997. 422 p. DOI: https://doi.org/10.1007/978-3-642-59237-9.
  8. Granero A. S., Moreno J. P., Phelps R. R. Mazur sets in normed spaces // Discrete Comput Geom. 2004. Vol. 31. P. 411–420. DOI: https://doi.org/10.1007/s00454-003-0808-5.
  9. Moreno J. P., Schneider R. Intersection properties of polyhedral norms // Adv. Geom. 2007. Vol. 7, № 3. P. 391–402. DOI: https://doi.org/10.1515/ADVGEOM.2007.025.
  10. Balashov M. V., Polovinkin E. S. M-strongly convex subsets and their generating sets. Sb. Math., 2000, vol. 191, no. 1, pp. 26–64. DOI: https://doi.org/10.1070/SM2000v191n01ABEH000447.
  11. Ivanov G. E. A criterion of smooth generating sets. Sb. Math., 2007, vol. 198, no. 3, pp. 343–368. DOI: https://doi.org/10.1070/SM2007v198n03ABEH003839.
  12. Balashov M. V., Ivanov G. E. Weakly convex and proximally smooth sets in Banach spaces. Izv. Math., 2009, vol. 73, no. 3, pp. 455–499. DOI: https://doi.org/10.1070/IM2009v073n03ABEH002454
  13. Schneider R. Convex bodies : The Brunn– Minkowski theory. Cambridge : Cambridge Univ. Press, 1993. DOI: https://doi.org/10.1017/CBO9780511526282.
Received: 
20.01.2016
Accepted: 
29.05.2016
Published: 
30.06.2016
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