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May 28, Wednesday
12:00 – 13:00

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. For an object $Q$ and a point $y$, let $mu_Q(y)$ be the average distance between $y$ and the points in $Q$, that is, $mu_Q(y)

int_{xin{Q}}left|{xy}right|dx/Area(Q)$, where $left|xyright|$ is the Euclidean distance between $x$ and $y$. Let $FWQ$ be a point for which this average distance is minimal, that is, $mu_Q(FWQ) = min_y mu_Q(y)$, and put $mu_Q^*

mu_Q(FWQ)$. The point $FWQ$ is a Fermat-Weber center of $Q$. I will 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 $4Delta(Q)/25$, and at most $2Delta(Q)/(3sqrt{3})$, where $Delta(Q)$ is the diameter of $Q$.