Spheres in infinite-dimensional normed spaces are Lipschitz contractible
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- by Y. Benyamini and Y. Sternfeld PDF
- Proc. Amer. Math. Soc. 88 (1983), 439-445 Request permission
Abstract:
Let $X$ be an infinite-dimensional normed space. We prove the following: (i) The unit sphere $\{ x \in X:\left \| x \right \| = 1\}$ is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of $X$ onto the unit sphere. (iii) There is a Lipschitz map $T$ of the unit ball into itself without an approximate fixed point, i.e. $\inf \{ \left \| {x - Tx} \right \|:\left \| x \right \| \leqslant 1\} > 0$.References
-
C. Bessaga and A. Pełczyński, Selected topics in infinite dimensional topology, PWN, Warsaw, 1975.
- Mahlon M. Day, Normed linear spaces, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973. MR 0344849
- K. Goebel, On the minimal displacement of points under Lipschitzian mappings, Pacific J. Math. 45 (1973), 151–163. MR 328708
- K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia Math. 47 (1973), 135–140. MR 336468, DOI 10.4064/sm-47-2-134-140
- 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
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 88 (1983), 439-445
- MSC: Primary 46B20; Secondary 57N17
- DOI: https://doi.org/10.1090/S0002-9939-1983-0699410-7
- MathSciNet review: 699410