Approximation theory and proximinal sets

Question

The question is to give an example such that the finite union of proximinal sets is not proximinal.

I have no idea to construct any example to suit this problem, will anybody help me?

Answer 1

There is no such example. Suppose $A_1,\dots,A_n$ are proximinal subsets of a normed space $X$, and $A=\bigcup_{i=1}^n A_i$. Let $x\in X$. For every $i$, pick $a_i\in A_i$ such that $\inf_{a\in A_i}\|x-a\| =\|x-a_i\|$. Among the points $a_i$, choose one that minimizes $\|x-a_i\|$. This point attains the minimum of $\|x-a\|$ over all $a\in A$.

Same works in metric spaces, actually.

It is true that the uniqueness of nearest point is not preserved under finite unions. That is, a finite union of Chebyshev sets is usually not Chebyshev. E.g., the union of two one-point sets.