Geodesic
Stability of the operator of $\varepsilon$-projection to the set of splines in~$C[0,1]$
Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 91-119

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We study the problem of the existence of a continuous selection for the metric projection to the set of $n$-link piecewise-linear functions in the space $C[0,1]$. We show that there is a continuous selection if and only if $n=1$ or $n=2$. We establish that there is a continuous $\varepsilon$-selection to $L$ ($L\subset C[0,1]$) if $L$ belongs to a certain class of sets that contains, in particular, the set of algebraic rational fractions and the set of piecewise-linear functions. We construct an example showing that sometimes there is no $\varepsilon$-selection for a set of splines of degree $d>1$. 
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     author = {E. D. Livshits},
     title = {Stability of the operator of $\varepsilon$-projection to the set of splines in~$C[0,1]$},
     journal = {Izvestiya. Mathematics },
     pages = {91--119},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2003_67_1_a5/}
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E. D. Livshits. Stability of the operator of $\varepsilon$-projection to the set of splines in~$C[0,1]$. Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 91-119. http://geodesic.mathdoc.fr/item/IM2_2003_67_1_a5/