Linearity of metric projections on Chebyshev subspaces in $L_1$ and $C$

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.