Selections of the best and near-best approximation operators and solarity

In a finite-dimensional Banach space, a closed set with lower semicontinuous metric projection is shown to have a continuous selection of the near-best approximation operator. Such a set is known to be a sun. In the converse question of the stability of best approximation by suns, it is proved that a strict sun in a finite-dimensional Banach space of dimension at most $3$ is a $P$-sun, has a contractible set of nearest points, and admits a continuous $\varepsilon $-selection from the operator of near-best approximation for any $\varepsilon >0$. A number of approximative and geometric properties of sets with lower semicontinuous metric projection are obtained.