New Criteria for the Existence of a Continuous $\varepsilon$-Selection

We study sets admitting a continuous selection of near-best approximations and characterize those sets in Banach spaces for which there exists a continuous $\varepsilon$-selection for each $\varepsilon>0$. The characterization is given in terms of $P$-cell-likeness and similar properties. In particular, we show that a closed uniqueness set in a uniformly convex space admits a continuous $\varepsilon$-selection for each $\varepsilon>0$ if and only if it is $\mathring{B}$-approximately trivial. We also obtain a fixed point theorem.