  The Fermat-Weber center of a planar body $Q$ is a point in the plane from
which the average distance to the points in $Q$ is minimal. We first show that
for any convex body $Q$ in the plane, the average distance from the
Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot
\Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture
of Carmi, Har-Peled and Katz. From the other direction, we prove that the same
average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490
\cdot \Delta(Q)$. The new bound substantially improves the previous bound of
$\frac{2}{3 \sqrt3} \cdot \Delta(Q) \approx 0.3849 \cdot \Delta(Q)$ due to
Abu-Affash and Katz, and brings us closer to the conjectured value of ${1/3}
\cdot \Delta(Q)$. We also confirm the upper bound conjecture for centrally
symmetric planar convex bodies.
