Geodesic
Finite-Dimensional Subspaces of~$L_p$ with Lipschitz Metric Projection
Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 514-525

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We prove that the metric projection onto a finite-dimensional subspace $Y\subset L_p$, $p\in(1,2)\cup(2,\infty)$, satisfies the Lipschitz condition if and only if every function in $Y$ is supported on finitely many atoms. We estimate the Lipschitz constant of such a projection for the case in which the subspace is one-dimensional.
Keywords: metric projection, Lipschitz condition, $L_p$ space, linearity coefficient. 
@article{MZM_2017_102_4_a3,
     author = {P. A. Borodin and Yu. Yu. Druzhinin and K. V. Chesnokova},
     title = {Finite-Dimensional {Subspaces} of~$L_p$ with {Lipschitz} {Metric} {Projection}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {514--525},
     publisher = {mathdoc},
     volume = {102},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a3/}
}
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P. A. Borodin; Yu. Yu. Druzhinin; K. V. Chesnokova. Finite-Dimensional Subspaces of~$L_p$ with Lipschitz Metric Projection. Matematičeskie zametki, Tome 102 (2017) no. 4, pp. 514-525. http://geodesic.mathdoc.fr/item/MZM_2017_102_4_a3/