Let $\mathcal{P}^n = \{ x \in \mathbb{R}^n : x \geq 0, \sum x = 1\}$.
Suppose I have $p_1, \ldots, p_m \in \mathcal{P}^n$. I want to find an $L^1$ center for these points. i.e. $q \in \mathcal{P}^n$ such that $\sum\limits_{j \leq n, k \leq m} |q_j - (p_k)_j|$ is minimized. Such a point obviously exists because $\mathcal{P}^n$ is compact, but I'd like to actually find one.
Any ideas how? I'm happy with an algorithm that would let me calculate the answer numerically as I don't really expect there's a closed form. Approximate answers are OK too.
Notes:
- If we drop the constraint that $\sum q = 1$ then letting $q_j$ be a median of $\{ (p_k)_j : k \leq m\}$ works, but in general this will not sum to 1.
- If we instead want the $L^2$ center then $\frac{1}{m} \sum p_j$ works, but this will generally not be an $L^1$ center.
- Any answer to this is presumably non-unique. It would be nice if there was some good "canonical" choice of center (e.g. one that minimizes the $L^2$ distance amongst the $L^1$ centers), but I don't really care that much.
- Answers which generalize to infinite measure spaces would be great, but I'm not holding out much hope.