Convexity of set $\{x\mid\operatorname{dist}(x,S) \leq \operatorname{dist}(x,T)\}$

Question

Is it possible to prove analytically the convexity of the set: $\{x\mid {\rm dist}(x,S) \leq {\rm dist}(x,T)\}$ where $S,T \subseteq \Bbb R^n$, and ${\rm dist}(x,S) = \inf\{\|x-z\|_2 \mid z \in S\}$?

Thanks in advance.

Answer 1

In general this set is not convex.

For example, in $\mathbb{R}$, $S =\{−1, 1\}, T = \{0\}$, we have $\{x\mid \operatorname{dist}(x,S)\le \operatorname{dist}(x,T)\}=\{x\in \mathbb{R}\mid x\le -\frac{1}{2} \text{ or } x\ge \frac{1}{2}\}$, which clearly is not convex.