Geodesic
The Lipschitz condition for a metric projection operator in the space $C[a,b]$
Matematičeskie zametki, Tome 22 (1977) no. 6, pp. 795-801

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It is proved that the metric projection operator onto a finite-dimensional Chebyshev subspace $M\subset C[a,b]$ locally uniformly satisfies the Lipschitz condition on the set $C[a,b]\setminus M$. 
@article{MZM_1977_22_6_a1,
     author = {A. V. Marinov},
     title = {The {Lipschitz} condition for a metric projection operator in the space $C[a,b]$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {795--801},
     publisher = {mathdoc},
     volume = {22},
     number = {6},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_6_a1/}
}
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A. V. Marinov. The Lipschitz condition for a metric projection operator in the space $C[a,b]$. Matematičeskie zametki, Tome 22 (1977) no. 6, pp. 795-801. http://geodesic.mathdoc.fr/item/MZM_1977_22_6_a1/