Solarity of boundedly weakly compact sets

We study boundedly weakly compact sets admitting, for each $\varepsilon>0$, $\mathrm{nw}$-continuous (norm-weak continuous) $\varepsilon$-selections of the near-best metric projection operator. Such a set in a reflexive Kadec–Klee space is shown to be a sun. For an approximatively compact set it is verified that this set admits an $\mathrm{nw}$-continuous $\varepsilon$-selection for each $\varepsilon>0$ if and only if it admits an $\mathrm{nn}$-continuous (norm-norm continuous) $\varepsilon$-selection for each $\varepsilon>0$.