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

In this paper for any epsilon > 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+epsilon, so that the Wosko constant W-gamma(C-m[0, 1]) is equal to 1