CONTINUITY OF ONE-SIDED BEST SIMULTANEOUS APPROXIMATIONS

  • Lee, Mun-Bae (DEPARTMENT OF MATHEMATICS, SOGANG UNIVERSITY) ;
  • Park, Sung-Ho (DEPARTMENT OF MATHEMATICS, SOGANG UNIVERSITY) ;
  • Rhee, Hyang-Joo (DEPARTMENT OF GENERAL STUDIES, DUKSUNG WOMEN’S UNIVERSITY)
  • Published : 2000.11.01

Abstract

In the space $C_1(X)$ of real-valued continuous functions with $L_1-norm$, every bounded set has a relative Chebyshev center in a finite-dimensional subspace S. Moreover, the set function $F\rightarrowZ_S(F)$ corresponding to F the set of its relative Chebyshev centers, in continuous on the space B[$C_1(X)$(X)] of nonempty bounded subsets of $C_1(X)$ (X) with the Hausdorff metric. In particular, every bounded set has a relative Chebyshev center in the closed convex set S(F) of S and the set function $F\rightarrowZ_S(F)$(F) is continuous on B[$C_1(X)$ (X)] with a condition that the sets S(.) are equal.

References

  1. Pacific Jour. Math. v.77 no.1 Chebyshev centers and uniform convexity P. Amir
  2. Doctoral Dissertation Parametric approximation S. G. Mabizela
  3. Bull. Korean Math. Soc. v.35 no.1 One-sided best simultaneous L₁approximation for a compact set S. H. Park;H. J. Rhee
  4. On L₁-approximation A. Pinkus
  5. Pacific Jour. Math. v.52 Chebyshev centers in spaces of continuous functions J. D. Ward
  6. Bull. Austral. Math. Soc. v.20 Best approximation and intersections of balls in Banach spaces D. Yost