A way to retract balls onto spheres (Q2720303)

Given an infinite-dimensional Banach space \(X\), denote by \(k(X)\) the smallest Lipschitz constant for a retraction of the unit ball of \(X\) onto its boundary. The finiteness of \(k(X)\) for any \(X\) has been proved by \textit{Y. Benyamini} and \textit{Y. Sternfeld} [Proc. Am. Math. Soc. 88, 439-445 (1983; Zbl 0518.46010)], and chapter of the book by the author and \textit{W. A. Kirk} [``Topics in metric fixed point theory' (1990; Zbl 0708.47031)] contains several upper estimates for specific spaces \(X\). In this paper the author discusses two methods for constructing such retractions rather explicitly, which makes it possible to sharpen \(k(X)\) for \(X= C[0,1]\) and \(X\) Hilbert. We remark that the corresponding problem for Lipschitz retractions replaced by \(\alpha\)-Lipschitz retractions, with \(\alpha\) being some measure of noncompactness, was considered (and in part sovled) by \textit{M. Väth} [Collect. Math. 52, No. 2, 101-116 (2001; Zbl 0996.47054)].