On the size of approximately convex sets in normed spaces

S. Dilworth; Ralph Howard; James Roberts

Studia Mathematica (2000)

  • Volume: 140, Issue: 3, page 213-241
  • ISSN: 0039-3223

Abstract

top
Let X be a normed space. A set A ⊆ X is approximately convexif 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 , C o ( A ) ) ≥ l o g 2 n - 1 and d i a m ( A ) ≤ C √ n ( l n 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.

How to cite

top

Dilworth, S., Howard, Ralph, and Roberts, James. "On the size of approximately convex sets in normed spaces." Studia Mathematica 140.3 (2000): 213-241. <http://eudml.org/doc/216765>.

@article{Dilworth2000,
abstract = {Let X be a normed space. A set A ⊆ X is approximately convexif 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. },
author = {Dilworth, S., Howard, Ralph, Roberts, James},
journal = {Studia Mathematica},
keywords = {approximately convex set; approximately convex function; convex hull; diameter; Hausdorff distance; space of type },
language = {eng},
number = {3},
pages = {213-241},
title = {On the size of approximately convex sets in normed spaces},
url = {http://eudml.org/doc/216765},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Dilworth, S.
AU - Howard, Ralph
AU - Roberts, James
TI - On the size of approximately convex sets in normed spaces
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 3
SP - 213
EP - 241
AB - Let X be a normed space. A set A ⊆ X is approximately convexif 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.
LA - eng
KW - approximately convex set; approximately convex function; convex hull; diameter; Hausdorff distance; space of type 
UR - http://eudml.org/doc/216765
ER -

References

top
  1. [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. Zbl0654.46018
  2. [2] R. E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, ibid. 34 (1981), 304-314. Zbl0475.47037
  3. [3] E. Casini and P. L. Papini, Almost convex sets and best approximation, Ricerche Mat. 40 (1991), 299-310. Zbl0758.41033
  4. [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. Zbl0840.26015
  5. [5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. Zbl0549.39006
  6. [6] J. Diestel and J. J. Uhl, Jr., Vector Measures, Amer. Math. Soc., Providence, RI, 1977. 
  7. [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. Zbl0949.26003
  8. [8] J. W. Green, Approximately subharmonic functions, Duke Math. J. 19 (1952), 499-504. Zbl0047.29601
  9. [9] D. H. Hyers, G. Isac and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, 1998. Zbl0907.39025
  10. [10] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821-828. Zbl0047.29505
  11. [11] M. Laczkovich, The local stability of convexity, affinity and of the Jensen equation, Aequationes Math., to appear. Zbl0934.39013
  12. [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. [13] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer, Berlin, 1991. Zbl0748.60004
  14. [14] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer, Berlin, 1977. Zbl0362.46013
  15. [15] I. P. Natanson, Theory of Functions of a Real Variable, Vol. 1, Ungar, New York, 1961. 
  16. [16] C. T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc. 118 (1993), 103-108. Zbl0823.26006
  17. [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. [18] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970. Zbl0193.18401
  19. [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. Zbl0759.46012

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.