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 $Q$.

One of the consequences of fast computers, the Internet and inexpensive storage is the widespread collection of data from a variety of sources and of a variety of types. Sources of data include web click streams, financial transactions, and observational science data. Data types include categorical vs. numerical, static vs. dynamic, points in a metric space vs. vertices in a graph. The nagging question often posed about these data sets is: Can we find something interesting that we did not already know? The first answer to this question is often: Let's try clustering the data! Indeed, clustering is one of the most widely used tools for analyzing data sets. Some modern applications of clustering include clustering the web, clustering search results, clustering click streams, customer segmentation, and community discovery in social networks.

Because of its recent ubiquitous applicability, the field of clustering has undergone major revolution over the last few decades characterized by advances in approximation and randomized algorithms, novel formulations of the clustering problem and algorithms for clustering data streams. This mini-course will cover some of these major advances particularly in the context of modern applications.