Abstract

We are concerned with some properties of the family of all subsets of a Banach space that can be written as an intersection of balls. A space with the Mazur Intersection Property (MIP) always satisfies those properties, so they can be regarded as weakenings of the MIP in different directions. The "ball hull" function (mapping a set to the intersection of all closed balls that cover it) is often an effective tool to study those properties.

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Keywords

  • Intersection of balls
  • Mazur intersection property
  • Mazur set
  • semidenting point
  • weak* denting point

Mathematics Subject Classification

46B20; 46B26, 60D05

References

  1. G. Beer: Topologies on Closed and Closed Convex Sets, Kluwer, Dordrecht (1993).https://doi.org/10.1007/978-94-015-8149-3Google Scholar
  2. V. G. Boltyanskij: Helly’s theorem for H-convex sets, Sov. Math., Dokl. 17 (1976) 78–81.Google Scholar
  3. J. Borwein, J. Vanderwerff: Dual Kadec-Klee norms and the relationships between Wijsman, slice and Mosco convergence, Mich. Math. J. 41 (1994) 371–387.https://doi.org/10.1307/mmj/1029005003Google Scholar
  4. D. Chen, B. L. Lin: On B-convex and Mazur sets of Banach spaces, Bull. Pol. Acad. Sci., Math. 43 (1995) 191–198.Google Scholar
  5. D. Chen, B. L. Lin: Ball separation properties in Banach spaces, Rocky Mt. J. Math. 28 (1998) 835–873.https://doi.org/10.1216/rmjm/1181071743Google Scholar
  6. M. Fabian: Review of [8], Math. Reviews MR2072538 2005h:46032 (2005).Google Scholar
  7. J. R. Giles, D. A. Gregory, B. Sims: Characterisation of normed linear spaces with Mazur’s intersection property, Bull. Aust. Math. Soc. 18 (1978) 105–123.https://doi.org/10.1017/S0004972700007863Google Scholar
  8. A. S. Granero, M. Jiménez-Sevilla, J. P. Moreno: Intersections of closed balls and geometry of Banach spaces, Extr. Math. 19 (2004) 55–92.Google Scholar
  9. A. S. Granero, J. P. Moreno, R. R. Phelps: Convex sets which are intersections of closed balls, Adv. Math. 183 (2004) 183–208.https://doi.org/10.1016/S0001-8708(03)00086-0Google Scholar
  10. A. S. Granero, J. P. Moreno, R. R. Phelps: Mazur sets in normed spaces, Discrete Comput. Geom. 31 (2004) 411–420.https://doi.org/10.1007/s00454-003-0808-5Google Scholar
  11. M. Jiménez Sevilla, J. P. Moreno: A note on porosity and the Mazur intersection property, Mathematika 47 (1997) 267–272.https://doi.org/10.1112/S0025579300015874Google Scholar
  12. S. Mazur: Über schwache Konvergenz in den Räumen Lp, Stud. Math. 4 (1933) 128–133.https://doi.org/10.4064/sm-4-1-128-133Google Scholar
  13. I. Molchanov: Theory of Random Sets, Springer, Berlin (2005).Google Scholar
  14. J. P. Moreno: Semicontinuous functions and convex sets in C(K) spaces, J. Aust. Math. Soc. 82 (2007) 111–121.https://doi.org/10.1017/S1446788700017493Google Scholar
  15. J. P. Moreno, P. L. Papini, R. R. Phelps: Diametrically maximal and constant width sets in Banach spaces, Cand. J. Math. 58 (2006) 820–842.https://doi.org/10.4153/CJM-2006-033-1Google Scholar
  16. J. P. Moreno, R. Schneider: Continuity properties of the ball hull mapping, Nonlinear Anal., Theory Methods Appl. 66A (2007) 914–925.https://doi.org/10.1016/j.na.2005.12.031Google Scholar
  17. R. R. Phelps: A representation theorem for bounded convex sets, Proc. Amer. Math. Soc. 11 (1960) 976–983.https://doi.org/10.1090/S0002-9939-1960-0123172-XGoogle Scholar
  18. H. Rådström: An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952) 165–169.https://doi.org/10.1090/S0002-9939-1952-0045938-2Google Scholar
  19. P. Terán: On the equivalence of Aumann and Herer expectations of random sets, Test 17 (2008) 505–514.https://doi.org/10.1007/s11749-007-0043-0Google Scholar
  20. P. Terán: Expectations of random sets, submitted.Google Scholar
  21. J. Vanderwerff: Mazur intersection properties for compact and weakly compact convex sets, Can. Math. Bull. 41 (1998) 225–230.https://doi.org/10.4153/CMB-1998-032-8Google Scholar

Suggested citation

P. Terán. “Intersections of Balls and the Ball Hull Mapping.” Journal of Convex Analysis 17 (2010), No. 1, 277–292. https://doi.org/10.68381/jca17020

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press.