Chebyshev unions of planes, and their approximative and geometric properties

We study approximative and geometric properties of Chebyshev sets composed of at most countably many planes (i.e., closed affine subspaces). We will assume that the union of planes is irreducible, i.e., no plane in this union contains another plane from the union. We show, in particular, that if a Chebyshev subset $M$ of a Banach space $X$ consists of at least two planes, then it is not $B$-connected (i.e., its intersection with some closed ball is disconnected) and is not $\mathring{B}$-complete. We also verify that, in reflexive $\left(\mathrm{CLUR}\right)$-spaces (and, in particularly, in complete uniformly convex spaces), a set composed of countably many planes is not a Chebyshev set. For finite unions, we show that any finite union of planes (involving at least two planes) is not a Chebyshev set for any norm on the space. Several applications of our results in the spaces $C\left(Q\right)$, $L^1$ and $L^{\infty }$ are also given.