On the Norm of the Metric Projections Original Research Article

LetXbe a Banach space. GivenMa subspace ofXwe denote withPMthe metric projection ontoM. We defineπ(X)≔ sup{‖PM‖: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=Lp, we obtain the inequality ‖PM‖⩽2|2/p−1|, for every subspaceMofLp. We also show thatπ(X)=π(X*).