Some of the classical triangle centers can be expressed as solutions to minimization problems: Given a triangle $A_1, A_2, A_3$ define $d_i, i=1,2,3$ to be the distance of a given point $P$ to $A_i$, and $f_q$ as the sum of the $q$-th power of these distances:$f_q = \sum_{i=1}^3 d_i^q$. I'm looking for the point $P$ which minimizes $f_q$. For $q=1$ this is the Steiner point, for $q=2$ the centroid, for $q \to \infty$ the circumcenter, for $q \to 0$ the point where the product of distances is minimized. An obvious question is to find the curve of all these points for reasonably general $q$ (e.g. $q \in \mathbb{R}_{>0}$). However, in the ressources for triangle centers, as e.g. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html this problem seems not to be considered.
EDIT:
1) I would like to restrict the problem to $q \ge 1$ since then $f_q$ is convex and a unique minimum is guaranteed.
2) I would like to add a generalization of the question: Consider all continuous functions $f(d_1,d_2,d_3)$ that a) are invariant under permutations of $d_1, d_2, d_3$ and b) have a unique minimum. What can be said about the locus of all these minima?
