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Spheres in infinite-dimensional normed spaces are Lipschitz contractible

Authors: Y. Benyamini and Y. Sternfeld
Journal: Proc. Amer. Math. Soc. 88 (1983), 439-445
MSC: Primary 46B20; Secondary 57N17
MathSciNet review: 699410
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Abstract: Let $ X$ be an infinite-dimensional normed space. We prove the following:

(i) The unit sphere $ \{ x \in X:\left\Vert x \right\Vert = 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\Vert {x - Tx} \right\Vert:\left\Vert x \right\Vert \leqslant 1\} > 0$.

References [Enhancements On Off] (What's this?)

  • [1] C. Bessaga and A. Pełczyński, Selected topics in infinite dimensional topology, PWN, Warsaw, 1975.
  • [2] M. M. Day, Normed linear spaces, 3rd ed., Springer-Verlag, Berlin, 1973. MR 0344849 (49:9588)
  • [3] K. Goebel, On the minimal displacement of points under Lipschitzian mappings, Pacific J. Math. 45 (1973), 151-163. MR 0328708 (48:7050)
  • [4] K. Goebel and W. A. Kirk, A fixed point theorem for transformations whose iterations have uniform Lipschitz constant, Studia Math. 47 (1973), 135-140. MR 0336468 (49:1242)
  • [5] 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. MR 616177 (82g:58008)

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Article copyright: © Copyright 1983 American Mathematical Society

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