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2000 | 140 | 3 | 213-241
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On the size of approximately convex sets in normed spaces

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EN
Abstrakty
EN
Let X be a normed space. A set A ⊆ X is approximately convex} if d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with $ℋ(A,Co(A))≥log_2n-1$ and $diam(A)≤C√n(ln n)^2$, where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.
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Twórcy
  • Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A., dilworth@math.sc.edu
autor
  • Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A., howard@math.sc.edu
  • Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A., roberts@math.sc.edu
Bibliografia
  • [1] J. Bourgain and S. J. Szarek, The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization, Israel J. Math. 62 (1988), 169-180.
  • [2] R. E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, ibid. 34 (1981), 304-314.
  • [3] E. Casini and P. L. Papini, Almost convex sets and best approximation, Ricerche Mat. 40 (1991), 299-310.
  • [4] E. Casini and P. L. Papini, A counterexample to the infinity version of the Hyers and Ulam stability theorem, Proc. Amer. Math. Soc. 118 (1993), 885-890.
  • [5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86.
  • [6] J. Diestel and J. J. Uhl, Jr., Vector Measures, Amer. Math. Soc., Providence, RI, 1977.
  • [7] S. J. Dilworth, R. Howard and J. W. Roberts, Extremal approximately convex functions and estimating the size of convex hulls, Adv. Math. 148 (1999), 1-43.
  • [8] J. W. Green, Approximately subharmonic functions, Duke Math. J. 19 (1952), 499-504.
  • [9] D. H. Hyers, G. Isac and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998.
  • [10] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821-828.
  • [11] M. Laczkovich, The local stability of convexity, affinity and of the Jensen equation, Aequationes Math., to appear.
  • [12] J.-O. Larsson, Studies in the geometrical theory of Banach spaces. Part 5: Almost convex sets in Banach spaces of type p,p>1, Ph.D. Thesis, Uppsala Univ., 1987.
  • [13] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, Berlin, 1991.
  • [14] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, Berlin, 1977.
  • [15] I. P. Natanson, Theory of Functions of a Real Variable, Vol. 1, Ungar, New York, 1961.
  • [16] C. T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc. 118 (1993), 103-108.
  • [17] G. Pisier, Sur les espaces qui ne contiennent pas de $ℓ_n^1$ uniformément, Séminaire Maurey-Schwartz 1973-74, École Polytechnique, Paris, 1974.
  • [18] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970.
  • [19] S. J. Szarek, On the geometry of the Banach-Mazur compactum, in: Functional Analysis (Austin, TX, 1987/1989), Lecture Notes in Math. 1470, Springer, Berlin, 1991, 48-59.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv140i3p213bwm
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