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2000 | 141 | 3 | 235-250
Tytuł artykułu

On Bárány's theorems of Carathéodory and Helly type

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Języki publikacji
EN
Abstrakty
EN
The paper begins with a self-contained and short development of Bárány's theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads as follows: if $C_{n}$, n=1,2,..., are families of closed convex sets in a bounded subset of a separable Banach space X such that there exists a positive $ε_{0}$ with $⋂_{C ∈ C_{n}} (C)_{ε} = ∅$ for $ε < ε_{0}$, then there are $C_{n} ∈ C_{n}$ with $⋂_{n} (C_{n})_{ε} = ∅$ for all $ε < ε_{0}$; here $(C)_{ε}$ denotes the collection of all x with distance at most ε to C.
Czasopismo
Rocznik
Tom
141
Numer
3
Strony
235-250
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-08-10
poprawiono
2000-04-11
Twórcy
  • I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 2-6, D-14195 Berlin, Germany
Bibliografia
  • [1] N. Alon and G. Kalai, Bounding the piercing number, Discrete Comput. Geom. 13 (1995), 245-256.
  • [2] M. Balaj and K. Nikodem, Remarks on Bárány's theorem and affine selections, preprint.
  • [3] I. Bárány, A generalization of Carathéodory's theorem, Discrete Math. 40 (1982), 141-152.
  • [4] I. Bárány, Carathéodory's theorem, colourful and applicable, in: Bolyai Soc. Math. Stud. 6, János Bolyai Math. Soc., 1997, 11-21.
  • [5] E. Behrends and K. Nikodem, A selection theorem of Helly type and its applications, Studia Math. 116 (1995), 43-48.
  • [6] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surveys Pure Appl. Math. 64, Longman Sci. Tech., 1993.
  • [7] J. Diestel and J. J. Uhl, Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
  • [8] J. Eckhoff, Helly, Radon, and Carathéodory type theorems, in: Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.), Elsevier, 1993, 389-448.
  • [9] R. C. James, A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. 80 (1974), 738-743.
  • [10] F. W. Levi, Eine Ergänzung zum Hellyschen Satze, Arch. Math. (Basel) 4 (1953), 222-224.
  • [11] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964).
  • [12] F. A. Valentine, Convex Sets, McGraw-Hill, 1964; reprinted by R. E. Krieger, 1976.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv141i3p235bwm
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