Geodesic
Linearity of metric projections on Chebyshev subspaces in $L_1$ and $C$
Matematičeskie zametki, Tome 63 (1998) no. 6, pp. 812-820.

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Let $Y$ be a Chebyshev subspace of a Banach space $X$. Then the single-valued metric projection operator $P_Y\colon X\to Y$ taking each $x\in X$ to the nearest element $y\in Y$ is well defined. Let $M$ be an arbitrary set, and let be a-finite measure on some $\sigma$-algebra $gS$ of subsets of $M$. We give a complete description of Chebyshev subspaces $Y\in L_1(M,\Sigma,\mu)$ for which the operator $P_Y$ is linear (for the space $L_1[0,1]$, this was done by Morris in 1980). We indicate a wide class of Chebyshev subspaces in $L_1(M,\Sigma,\mu)$, for which the operator $P_Y$ is nonlinear in general. We also prove that the operator $P_Y$, where $Y\subset C[K]$ is a nontrivial Chebyshev subspace and $K$ is a compactum, is linear if and only if the codimension of $Y$ in $C[K]$ is equal to 1. 
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     author = {P. A. Borodin},
     title = {Linearity of metric projections on {Chebyshev} subspaces in $L_1$ and $C$},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
     volume = {63},
     number = {6},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_63_6_a1/}
}
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P. A. Borodin. Linearity of metric projections on Chebyshev subspaces in $L_1$ and $C$. Matematičeskie zametki, Tome 63 (1998) no. 6, pp. 812-820. http://geodesic.mathdoc.fr/item/MZM_1998_63_6_a1/

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