Abstract
We obtain formulae to calculate the asymptotic center and radius of bounded sequences in \({\mathcal {C}}_0(L)\) spaces. We also study the existence of continuous selectors for the asymptotic center map in general Banach spaces. In Hilbert spaces, even a Hölder-type estimation is given.
Similar content being viewed by others
References
Amir, D., Mach, J., Saatkamp, K.: Existence of Chebyshev centers, best \(n\)-nets and best compact approximants. Trans. AMS 271(2), 513–524 (1982)
Amir, D., Mach, J.: Chebyshev centers in normed spaces. J. Approx. Theory 40, 364–374 (1984)
Baronti, M., Papini, P.L.: “Nearby sets and centers”, approximation and optimization (Havana, 1987). Lecture notes in mathematics, vol. 1354, pp. 98–105 (1988)
Edelstein, M.: The construction of an asymptotic center with a fixed-point property. Bull. Am. Math. Soc. 78, 206–208 (1972)
Garkavi, A.L.: On the optimal net and best cross-section of a set in a normed space. Izv. Akad. Nauk SSSR Ser. Mat. 26, 87–106 (1962). (Russian)
Lim, T.C.: A fixed point theorem for families on nonexpansive mappings. Pac. J. Math. 53, 487–493 (1974)
Lim, T.C.: Asymptotic centers and nonexpansive mappings in conjugate Banach spaces. Pac. J. Math. 90(1), 135–143 (1980)
Lim, T.C.: Asymptotic centers in \(c_0\), \(c\) and \(m\). Contemp. Math. 18, 141–154 (1983)
Listán-García, M.C., Rambla-Barreno, F.: Rough convergence and Chebyshev centers in Banach spaces. Numer. Funct. Anal. Optim. 35(4), 432–442 (2014)
Mach, J.: Continuity properties of Chebyshev centers. J. Approx. Theory 29, 223–230 (1980)
Michael, E.: Continuous selections. I. Ann. Math. 63(2), 361–382 (1956)
Nagata, J.: Modern General Topology, 2nd edn. North-Holland, Amsterdam (1985)
Veselý, L.: Chebyshev centers in hyperplanes of \(c_0\). Czechoslovak Math. J. 52(4), 721–729 (2002)
Zamjatin, V.N.: The Chebyshev center in hyperspaces of continuous functions. In: Štraus A.V. (ed.) Funktsional’nyj Analiz, vol. 12, pp. 56–68. Ul’janovsk. Gos. Ped. Inst., Ul’janovsk (1979) (Russian)
Acknowledgments
The authors would like to thank Bernardo Cascales for his many suggestions that have greatly improved this paper. Our gratitude is also due to the referees for several helpful remarks. Finally, the second and third author would like to thank Carlos Angosto and Bernardo Cascales for their warm hospitality during a short stay at University of Murcia.
Author information
Authors and Affiliations
Corresponding author
Additional information
C. Angosto was partially supported by the MINECO/FEDER project MTM2014-57838-C2-1-P. M. C. Listán-García and F. Rambla-Barreno authors were partially supported by Junta de Andalucía and FEDER Grant FQM-257.
Rights and permissions
About this article
Cite this article
Angosto, C., Listán-García, M.C. & Rambla-Barreno, F. Continuity properties of sequentially asymptotically center-complete spaces. RACSAM 110, 809–822 (2016). https://doi.org/10.1007/s13398-015-0268-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-015-0268-9