We show that for any convex object Q in the plane, the average
distance from the Fermat-Weber center of Q to the points in
Q is at least \Diam(P)/7, where \Diam(P) is the
diameter of P, and that there exists a convex object for which
this distance is \Diam(P)/6. We use this result to obtain a
linear-time approximation scheme for finding an approximate
Fermat-Weber center of a convex polygon Q.

@article{DBLP:journals/comgeo/CarmiHK05,
author = {Paz Carmi and
Sariel Har-Peled and
Matthew J. Katz},
title = {On the Fermat-Weber center of a convex object.},
journal = {Comput. Geom.},
volume = {32},
number = {3},
year = {2005},
pages = {188-195},
ee = {http://dx.doi.org/10.1016/j.comgeo.2005.01.002},
bibsource = {DBLP, http://dblp.uni-trier.de}
}