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. 
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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/

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