Kopecká, Eva; Reich, Simeon Nonexpansive retracts in Banach spaces. (English) Zbl 1125.46019 Jachymski, Jacek (ed.) et al., Fixed point theory and its applications. Proceedings of the international conference, Bȩdlewo, Poland, August 1–5, 2005. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 77, 161-174 (2007). A nonexpansive retract of a Banach space or a metric space is the range of a nonexpansive retraction. A space has the compact nonexpansive envelope property if each compact subset is contained in a compact nonexpansive retract. It is shown that \(c_0\) has this property while \(\ell_p\) lacks it when \(1\leq p<\infty\) and \(p\neq2\); other related examples and counterexamples are given. It is also shown that the unit ball of \(\ell_1\) is not a nonexpansive retract (for \(\ell_p\), \(1<p<\infty\), \(p\neq2\), this follows from a result of R. E. Bruck [Proc. Am. Math. Soc. 43, 173–175 (1974; Zbl 0306.46027)]). Finally, it is shown that in many Banach spaces the nonexpansive retracts are even the range of a “sunny” nonexpansive retraction \(R\), i.e.,\(R(v+r(x-v))=v\) whenever \(Rx=v\) and \(r\geq0\).For the entire collection see [Zbl 1112.47302]. Reviewer: Martin Väth (Gießen) Cited in 47 Documents MSC: 46B99 Normed linear spaces and Banach spaces; Banach lattices 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 52A05 Convex sets without dimension restrictions (aspects of convex geometry) 52A55 Spherical and hyperbolic convexity Keywords:Banach space; nonexpansive retraction; compact nonexpansive envelope property; sunny nonexpansive retraction Citations:Zbl 0306.46027 PDFBibTeX XMLCite \textit{E. Kopecká} and \textit{S. Reich}, Banach Cent. Publ. 77, 161--174 (2007; Zbl 1125.46019) Full Text: Link