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 $$\varepsilon>0$$ .