Ambiguous loci of the farthest distance mapping from compact convex sets

F. De Blasi; J. Myjak

Studia Mathematica (1995)

  • Volume: 112, Issue: 2, page 99-107
  • ISSN: 0039-3223

Abstract

top
Let be a strictly convex separable Banach space of dimension at least 2. Let K() be the space of all nonempty compact convex subsets of endowed with the Hausdorff distance. Denote by K 0 the set of all X ∈ K() such that the farthest distance mapping a ↦ M X ( a ) is multivalued on a dense subset of . It is proved that K 0 is a residual dense subset of K().

How to cite

top

De Blasi, F., and Myjak, J.. "Ambiguous loci of the farthest distance mapping from compact convex sets." Studia Mathematica 112.2 (1995): 99-107. <http://eudml.org/doc/216147>.

@article{DeBlasi1995,
abstract = {Let be a strictly convex separable Banach space of dimension at least 2. Let K() be the space of all nonempty compact convex subsets of endowed with the Hausdorff distance. Denote by $K^0$ the set of all X ∈ K() such that the farthest distance mapping $a ↦ M_X(a)$ is multivalued on a dense subset of . It is proved that $K^0$ is a residual dense subset of K().},
author = {De Blasi, F., Myjak, J.},
journal = {Studia Mathematica},
keywords = {convex sets; farthest points; distance mapping},
language = {eng},
number = {2},
pages = {99-107},
title = {Ambiguous loci of the farthest distance mapping from compact convex sets},
url = {http://eudml.org/doc/216147},
volume = {112},
year = {1995},
}

TY - JOUR
AU - De Blasi, F.
AU - Myjak, J.
TI - Ambiguous loci of the farthest distance mapping from compact convex sets
JO - Studia Mathematica
PY - 1995
VL - 112
IS - 2
SP - 99
EP - 107
AB - Let be a strictly convex separable Banach space of dimension at least 2. Let K() be the space of all nonempty compact convex subsets of endowed with the Hausdorff distance. Denote by $K^0$ the set of all X ∈ K() such that the farthest distance mapping $a ↦ M_X(a)$ is multivalued on a dense subset of . It is proved that $K^0$ is a residual dense subset of K().
LA - eng
KW - convex sets; farthest points; distance mapping
UR - http://eudml.org/doc/216147
ER -

References

top
  1. [1] E. Asplund, Farthest points in reflexive locally uniformly rotund Banach spaces, Israel J. Math. 4 (1966), 213-216. Zbl0143.34904
  2. [2] K. Bartke and H. Berens, Eine Beschreibung der Nichteindeutigkeitsmenge für die beste Approximation in der Euklidischen Ebene, J. Approx. Theory 47 (1986), 54-74. Zbl0619.41020
  3. [3] J. M. Borwein and S. Fitzpatrick, Existence of nearest points in Banach spaces, Canad. J. Math. 41 (1989), 702-720. Zbl0668.46006
  4. [4] F. S. De Blasi, J. Myjak and P. L. Papini, Porous sets in best approximation theory, J. London Math. Soc. (2) 44 (1991), 135-142. Zbl0786.41027
  5. [5] R. Deville and V. Zizler, Farthest points in w*-compact sets, Bull. Austral. Math. Soc. 38 (1988), 433-439. Zbl0656.46012
  6. [6] N. Dunford and J. T. Schwartz, Linear Operators, Vol. I, Interscience, New York, 1958. Zbl0084.10402
  7. [7] M. Edelstein, Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171-176. Zbl0151.17601
  8. [8] P. M. Gruber, Die meisten konvexen Körper sind glatt, aber nicht zu glatt, Math. Ann. 229 (1977), 259-266. Zbl0342.52009
  9. [9] K. S. Lau, Farthest points in weakly compact sets, Israel J. Math. 22 (1975), 168-174. Zbl0325.46022
  10. [10] V. Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51-63. Zbl0092.11602
  11. [11] S. Miyajima and F. Wada, Uniqueness of a farthest point in a bounded closed set in Banach spaces, preprint. Zbl0798.46009
  12. [12] B. B. Panda and K. Dwivedi, On existence of farthest points, Indian J. Pure Appl. Math. 16 (1985), 486-490. Zbl0584.46008
  13. [13] T. Zajíček, On the Fréchet differentiability of distance functions, Rend. Circ. Mat. Palermo (2) Suppl. 5 (1985), 161-165. Zbl0581.41028
  14. [14] T. Zamfirescu, Using Baire category in geometry, Rend. Sem. Mat. Univers. Politec. Torino 43 (1985), 67-88. Zbl0601.52004
  15. [15] T. Zamfirescu, The nearest point mapping is single valued nearly everywhere, Arch. Math. (Basel) 54 (1990), 563-566. Zbl0715.54013

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.