Minimal displacement and retraction problems in infinite-dimensional Hilbert spaces

Bolibok, Krzysztof

doi:10.1090/S0002-9939-03-07150-8

Minimal displacement and retraction problems in infinite-dimensional Hilbert spaces
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Proc. Amer. Math. Soc. 132 (2004), 1103-1111 Request permission

Abstract:

We give the first constructive example of a Lipschitz mapping with positive minimal displacement in an infinite-dimensional Hilbert space $H.$ We use this construction to obtain an evaluation from below of the minimal displacement characteristic in the space $H.$ In the second part we present a simple and constructive proof of existence of a Lipschitz retraction from a unit ball $B$ onto a unit sphere $S$ in the space $H$, and we improve an evaluation from above of a retraction constant $k_{0}\left ( H\right ) .$

References

Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), no. 3, 439–445. MR 699410, DOI 10.1090/S0002-9939-1983-0699410-7
Krzysztof Bolibok, Constructions of Lipschitzian mappings with non-zero minimal displacement in spaces $L^1(0,1)$ nd $L^2(0,1)$, Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 25–31. MR 1472574
Krzysztof Bolibok, Construction of a Lipschitzian retraction in the space $c_0$, Proceedings of Workshop on Fixed Point Theory (Kazimierz Dolny, 1997), 1997, pp. 43–46. MR 1666165
Krzysztof Bolibok and Kazimierz Goebel, A note on minimal displacement and retraction problems, J. Math. Anal. Appl. 206 (1997), no. 1, 308–314. MR 1429293, DOI 10.1006/jmaa.1997.5167
C. Franchetti, Lipschitz maps and the geometry of the unit ball in normed spaces, Arch. Math. (Basel) 46 (1986), no. 1, 76–84. MR 829819, DOI 10.1007/BF01197144
M. Furi and M. Martelli, On the minimal displacement of points under $\alpha$-Lipschitz maps in normed spaces, Boll. Un. Mat. Ital. (4) 9 (1974), 791–799 (English, with Italian summary). MR 0370282
K. Goebel, On the minimal displacement of points under Lipschitzian mappings, Pacific J. Math. 45 (1973), 151–163. MR 328708, DOI 10.2140/pjm.1973.45.151
Kazimierz Goebel, Metric environment of the topological fixed point theorems, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 577–611. MR 1904288
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
Kazimierz Goebel and Tomasz Komorowski, Retracting balls onto spheres and minimal displacement problem, Fixed point theory and applications (Marseille, 1989) Pitman Res. Notes Math. Ser., vol. 252, Longman Sci. Tech., Harlow, 1991, pp. 155–172 (English, with French summary). MR 1122826
Tomasz Komorowski and Jacek Wośko, A remark on the retracting of a ball onto a sphere in an infinite-dimensional Hilbert space, Math. Scand. 67 (1990), no. 2, 223–226. MR 1096458, DOI 10.7146/math.scand.a-12334
P. K. Lin and Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), no. 4, 633–639. MR 776193, DOI 10.1090/S0002-9939-1985-0776193-5
Bogdan 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), no. 11-12, 861–864 (1981) (English, with Russian summary). MR 616177
Wilhelm Wirtinger, Über eine Minimalaufgabe im Gebiete der analytischen Funktionen von mehreren Veränderlichen, Monatsh. Math. Phys. 47 (1939), 426–431 (German). MR 56, DOI 10.1007/BF01695512
Simeon Reich, Minimal displacement of points under weakly inward pseudo-Lipschitzian mappings. II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 60 (1976), no. 2, 95–96 (English, with Italian summary). MR 487647
R. L. Thele, Some results on the radial projection in Banach spaces, Proc. Amer. Math. Soc. 42 (1974), 483–486. MR 328550, DOI 10.1090/S0002-9939-1974-0328550-1