Geodesic
The linearity of metric projection operator for subspaces of $L_p$ spaces
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2010), pp. 30-36

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Let $Y$ be a Chebyshev subspace of a Banach space $X$. Then the single-valued metric projection operator $P_Y:X\to Y$ taking each $x\in X$ to the nearest element $y\in Y$ is well defined. Let $M$ be an arbitrary set and $\mu$ be a $\sigma$-finite measure on some $\sigma$-algebra $\Sigma$ of subsets of $M$. We give a description of Chebyshev subspaces $Y\subset L_p(M,\Sigma,\mu)$ with finite dimension and finite codimension the operator $P_Y$ is linear for. 
@article{VMUMM_2010_1_a4,
     author = {Yu. Yu. Druzhinin},
     title = {The linearity of metric projection operator for subspaces of $L_p$ spaces},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {30--36},
     publisher = {mathdoc},
     number = {1},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2010_1_a4/}
}
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Yu. Yu. Druzhinin. The linearity of metric projection operator for subspaces of $L_p$ spaces. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2010), pp. 30-36. http://geodesic.mathdoc.fr/item/VMUMM_2010_1_a4/