Bounded contractibility of strict suns in three-dimensional spaces

A strict sun in a finite-dimensional (asymmetric) normed space $X$, $\operatorname {dim}X \le 3$, is shown to be $P$-contractible, $P$-solar, $\mathring B $-infinitely connected, $\mathring B $-contractible, $\mathring B $-retract, and having a continuous additive (multiplicative) $\varepsilon$-selection for any $\varepsilon > 0$. A $P$-acyclic subset of a three-dimensional space is shown to have a continuous $\varepsilon$-selection for any $\varepsilon > 0$. For the dimension $3$ the well-known Tsar'kov's characterization of spaces, in which any bounded Chebyshev set is convex, is extended to the case of strict suns.