My overall goal is to place $k$ "central" points in a region such that the sum of distance from other points to these points is minimized.
Let me start with the discrete question.
Suppose there are $n$ points $p_1, \dots ,p_n$ given in the 2 dimensional plane with co-ordinates $p_i = (x_i, y_i)$. For a given integer $k$, you need to place $k$ points (need not be one of the original points) with $q_j = (a_j, b_j)$ such that the following is minimized $$ \sum_{i = 1}^n \sum_{j = 1}^k d(p_i, q_j) $$ where $d(\cdot, \cdot)$ is a distance metric. For example, we could use the Eucledian distance i.e. $d(p_i, q_j) = \sqrt{(x-a)^2 + (y-b)^2}$
Note that when $k = 1$, this is exactly the problem of finding a 2-d geometric median. Of course, the problem can be stated for higher dimensions as well and/or generalized to Reimannian manifolds, but for now, two dimensions is good enough.
An extension that I am interested in is for a continuous region. Specifically,
Given a continuous region $\mathcal{R} \subset \mathbb{R}^n$ and a positive integer $k$, place $k$ points (in $\mathcal{R}^n$) defined by $$ \arg \min_{q_1,\dots ,q_k} \int_\mathcal{p \in R} \sum_{j=1}^k d(p,q_j) $$ where $d(\cdot, \cdot)$ is a distance function (e.g. L2 norm $d(p,q) = {||p-q||}_2$)
For both questions, the goal is find an algorithm to compute the $k$ points. There likely is not an explicit formula (since there isn't one for the geometric median either), so an iterative algorithm should work perfectly.
[Edit] A problem related to the discrete one seems to be the $k$-medians problem. Here, given $n$ points, the goal is to chose $k$ of them as medians, such that the sum of distances of each of the $n$ points to their closest median is minimized.