TITLE: Algorithm for k-medians in a convex polygon
QUESTION [6 upvotes]: Are there any known approximation algorithms or exact solution schemes for the k-medians problem in a convex polygon?  That is, placing a collection of points $p_1,\dots,p_k \subset \mathbb{R}^2$ in a convex polygon $C$ so as to minimize $$\iint_C \min_i \|x-p_i\| dx$$

REPLY [4 votes]: A while ago I wrote, but never published, an approximation algorithm for this problem.  Using some new results and updating the citations, it looks like I can get the approximation constant down to 9.026 (assuming I didn't make any mistakes).  It's not clear to me if that's publication-worthy, but I uploaded a draft to
http://www.tc.umn.edu/~jcarlsso/fermat-weber.pdf
if anyone is interested.