Remarks on Differentiability of Metric Projections onto Cones of Nonnegative Functions

The differentiability properties of the metric projections onto the cones of nonnegative functions in $L^p(0,1)$ and in $W^{1,p}(0,1)$ are considered. It is shown that the metric projection mapping is Bouligand differentiable in $L^p(0,1)$, but it is not Bouligand differentiable in $W^{1,p}(0,1)$.