Given a set of points $S \subset X$ with a metric defined $d : X \times X \to \mathbb{R}^+$.
I am interested in the relationship between $Z = \textbf{argmin}_{z \in X} \sum_{i \in S} d(i, z)$ and $S$.
Here $z \in X$ is the global minimum distance element to $S$, as it is not necessarily unique, I use $Z$ to capture all the possible $z$.
Under what condition would $Z \cap S = \emptyset$ ? Also under what condition $Z \cap S \neq \emptyset$ ?
For example, if $X$ is the $\mathbb{R}$, and $d$ is the euclidean distance then $Z \cap S \neq \emptyset$. As $Z \cap S$ is simply the median of $S$.
Or if $X$ is $\mathbb{R}^2$ and $d$ is still the euclidean distance, assuming no 3 elements in $S$ is co-linear, then $Z \cap S = \emptyset$. If only $S$ only contains 3 non co-linear element, the $Z$ is Fermat–Torricelli point.