Abstract
We answer in the affirmative the following question raised by H. H. Corson in 1961: “Is it possible to cover every Banach space X by bounded convex sets with non-empty interior in such a way that no point of X belongs to infinitely many of them?”
Actually, we show the way to produce in every Banach space X a bounded convex tiling of order 2, i.e., a covering of X by bounded convex closed sets with non-empty interior (tiles) such that the interiors are pairwise disjoint and no point of X belongs to more than two tiles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. H. Corson, Collections of convex sets which cover a Banach space, Fundamenta Mathematicae 49 (1961), 143–145.
V. P. Fonf, Three characterizations of polyhedral Banach spaces, Akademiya Nauk Ukrainskoĭ SSR. Institut Matematiki. Ukrainskiĭ Matematicheskiĭ Zhurnal 42 (1990), 1286–1290.
V. P. Fonf and J. Lindenstrauss, Some results on infinite-dimensional convexity, Israel Journal of Mathematics 108 (1998), 13–32.
V. P. Fonf, A. Pezzotta and C. Zanco, Tiling infinite-dimensional normed spaces, The Bulletin of the London Mathematical Society 29 (1997), 713–719.
V. P. Fonf and C. Zanco, Covering a Banach space, Proceedings of the American Mathematical Society 134 (2006), 2607–2611.
V. P. Fonf and C. Zanco, Finitely locally finite coverings of Banach spaces, Journal of Mathematical Analysis and Applications 350 (2009), 640–650.
V. P. Fonf and C. Zanco, Coverings of Banach spaces: beyond the Corson theorem, Forum Mathematicum 21 (2009), 533–546.
W. B. Johnson and J. Lindenstrauss, Basic Concepts in the Geometry of Banach Spaces, in Handbook of the Geometry of Banach Spaces, Volume 1, (W. B. Johnson and J. Lindenstrauss, eds.), North-Holland Elsevier, Amsterdam, 2001.
V. Klee, Dispersed Chebyshev sets and coverings by balls, Mathematische Annalen 257 (1981), 251–260.
V. Klee, Do infinite-dimensional Banach spaces admit nice tilings?, Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences 21 (1986), 415–427.
J. Lindenstrauss, Weakly compact sets, their topological properties and the Banach spaces they generate, in Symposium on Infinite-Dimensional Topology, Annals of Mathematics Studies 69 (R. D. Anderson, ed.), Princeton University Press, Princeton, NJ, 1973, pp. 235–273.
M. J. Nielsen, Singular points of a convex tiling, Mathematische Annalen 284 (1989), 601–616.
A. H. Stone, Closed tiling of Euclidean spaces, Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento No. 8 (1985), 321–324.
C. Zanco, Even infinite-dimensional Banach spaces can enjoy carpeting and tiling, in Proceedings of the 13th Seminar on Analysis and Its Applications, Isfahan University Press, Iran, 2003, pp. 121–141.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of the first author was supported by the GNAMPA of the Istituto Nazionale di Alta Matematica of Italy.
Research of the second author was supported in part by the GNAMPA of the Istituto Nazionale di Alta Matematica of Italy and in part by the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev, Beer-Sheva, Israel.
Rights and permissions
About this article
Cite this article
Marchese, A., Zanco, C. On a question by Corson about point-finite coverings. Isr. J. Math. 189, 55–63 (2012). https://doi.org/10.1007/s11856-011-0126-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-011-0126-1