The Linearity Coefficient of the Metric Projection onto a Chebyshev Subspace

The linearity coefficient $\lambda(Y)$ of a metric projection $P_Y$ onto a subspace $Y$ in a Banach space $X$ is determined. This coefficient turns out to be related to the Lipschitz norm of the operator $P_Y$. It is proved that, for any Chebyshev subspace $Y$ in the space $C$ or $L_1$, either $\lambda(Y)=1$ (which corresponds to the linearity of $P_Y$) or $\lambda(Y)\le 1/2$.