Abstract: The Fermat-Weber center of an object $Q$ in the plane
is a point in the plane such that the average distance from it to
the points in $Q$ is minimal. In this work, we study the relation between
the minimal average distance and the diameter of the object. We show that
for any convex object $Q$ in the plane, the average distance between
the Fermat-Weber center of $Q$ and the points in $Q$ is at least
$4\Delta(Q)/25$ and at most $2\Delta(Q)/(3\sqrt{3})$, where $\Delta(Q)$ is
the diameter of $Q$.
We generalize these results to higher dimensions and show that
for any convex object $Q$ in $\mathbb{R}^d$, the average distance between
the Fermat-Weber center of $Q$ and the points in $Q$ is
at least $2^d\Delta(Q)/(3^{d+1}-2)$ and at most $\sqrt{d^3\Delta(Q)/(2(d+1)^3)}$.