Lipschitz retractions onto sphere vs spherical cup in a Hilbert space

Phichet Chaoha, Jumpot Intrakul, Wacharin Wichiramala



We prove that, in every infinite dimensional Hilbert space, there exists $t_0> -1$ such that the smallest Lipscthiz constant of retractions from the unit ball onto its boundary is the same as the smallest Lipschitz constant of retractions from the unit ball onto its $t$-spherical cup for all $t\in[-1,t_0]$.


Lipschitz retraction; optimal retraction; spherical cup

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M. Annoni and E. Casini, An upper bound for the Lipschitz retraction constant in l1, Studia. Math. 180 (2007) no. 1, 73–76.

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

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

K. Bolibok, Minimal displacement and retraction problems in infinite-dimensional Hilbert spaces, Proc. Amer. Math. Soc. 132 (2003), 1103–1111.

E. Casini and L. Piasecki, The minimal displacement and optimal retraction problems in some Banach spaces, J. Nonlinear Convex Anal. 18 (2017), no. 1, 61–71.

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

P. Chaoha and J. Intrakul, Retraction from a unit ball onto its spherical cup, Linear Nonlinear Anal. 2 (2016), no. 1, 17–28.

K. Goebel and W.A. Kirk, Topics in Metric FixedPpoint Theory, Cambridge University Press, London, 1990.

T. Komorowski and J. Wośko, A remark on the retracting of a ball onto a sphere in an infinite dimensional Hilbert space, Math. Scand. 67 (1990), 223–226.

B. Nowak, On the Lipschitzian retraction of the unit ball in infinite dimensional Banach spaces onto its boundary, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 861–864.

L. Piasecki, Retracting ball onto sphere in BC0 (R), Topol. Methods Nonlinear Anal. 33 (2009), no. 2, 307–314.

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


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