The linearity of metric projection operator for subspaces of $L_p$ spaces

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.