Optimal retraction problem for proper $k$-ball-contractive mappings in $C^m [0,1

Abstract

In this paper for any $\eps > 0$ we construct a new proper $k$-ball-contractive retraction of the closed unit ball of the Banach space $C^m [0,1]$ onto its boundary with $k < 1+ \eps$, so that the Wo\'sko constant $W_\gamma (C^m [0,1])$ is equal to $1$.