Openness of the metric projection in certain Banach spaces (Q1062224)

Let E be a Banach space and \(K\subset E\) a nonempty closed convex subset containing \(\theta\) as an interior point. The metric projection \(P_ K\) is said to be open [weakly open] provided the image of each open [weakly open] subset of \(E\setminus K\) is a relatively norm-open subset of the boundary of K. Set \(\mu_ K(x)=\inf \{\lambda >0;x\in \lambda K\}\), \(x\in E\). Relations between the openness [the weakly openness] of \(P_ K\) and the Fréchet [Gateaux] differentiability of \(\mu_ K\) at the points of the boundary of K are proved, under different assumptions on the geometry of E.