The linearity coefficient of metric projections onto a Chebyshev subspace

The linearity coefficient λ(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 λ(Y) = 1 (which corresponds to the linearity of P _Y ) or λ(Y) ≤ 1/2.