Definition: The distance between two points $p_1, p_2 \in \mathbb{R}^n$ is
\begin{equation}
d(p_1,p_2)=|| p_1-p_2 ||.
\end{equation}
The distance between a point $p\in \mathbb{R}^n$ and a set $\mathcal{C} \subset \mathbb{R}^n$ is
\begin{equation}
d(p,\mathcal{C}) = \inf\{||p-q|| : q \in \mathcal{C}\}.
\end{equation}
Given three points $p_i \in \mathbb{R}^n, i=1,2,3$. The triangle inequality tells us that \begin{equation} |d(p_1,p_3) - d(p_3,p_2)| \le d(p_1, p_2) \le d(p_1, p_3) + d(p_3,p_2). \end{equation}
Now the problem is:
Is the triangle inequality still held if one point is replaced by a non-singleton set?