Remarks on retracting balls on spherical caps in \(c_{0}\), \(c\), \(l^{\infty }\) spaces

Kazimierz Goebel

Abstract


For any infinite dimensional Banach space there exists a lipschitzian retraction of the closed unit ball B onto the unit sphere S. Lipschitz constants for such retractions are, in general, only roughly estimated. The paper is illustrative. It contains remarks, illustrations and estimates concerning optimal retractions onto spherical caps for sequence spaces with the uniform norm.

Keywords


Retraction; Lipschitz constant; radial projection; truncation; spherical cap

Full Text:

PDF

References


Annoni, M., Casini, E., An upper bound for the Lipschitz retraction constant in l_1, Studia Math. 180 (2007), 73–76.

Baronti, M., Casini, E., Franchetti, C., The retraction constant in some Banach spaces, J. Approx. Theory 120 (2) (2003), 296–308.

Benyamini, Y., Sternfeld, Y., Spheres in infinite dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439–445.

Casini, E., Piasecki, Ł, The minimal displacement and optimal retraction problems in some Banach spaces, J. Nonlinear Convex Anal. 18 (1) (2017), 61–71.

Chaoha, P., Goebel, K., Termwuttipong, I., Around Ulam question on retractions, Topol. Methods Nonlinear Anal. 40 (2012), 215–224.

Chaoha, P., Intracul, J., Wichramala, W., Lipschitz retractions onto sphere vs spherical cup, Topol. Methods Nonlinear Anal. 52 (2) (2018), 677–691.

Goebel, K., Concise Course of Fixed Point Theorems, Yokohama Publishers, Yokohama, 2002.

Goebel, K., Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.

Goebel, K., Marino, G., Muglia, L., Volpe, R., The retraction constant and minimal displacement characteristic of some Banach spaces, Nonlinear Anal. 67 (2007), 735–744.

Kirk, W. A., Sims, B. (eds.), Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers, Dordrecht, 2001.

Piasecki, Ł, Retracting ball onto sphere in some Banach spaces, Nonlinear Anal. 74 (2) (2011), 396–399.

Piasecki, Ł, Retracting ball onto sphere in BC_0 (R), Topol. Methods Nonlinear Anal. 33 (2) (2009), 307–314.




DOI: http://dx.doi.org/10.17951/a.2020.74.1.45-55
Date of publication: 2020-10-20 20:08:04
Date of submission: 2020-10-11 14:18:58


Statistics


Total abstract view - 756
Downloads (from 2020-06-17) - PDF - 500

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2020 Kazimierz Goebel Goebel