  We consider the continuous Fermat-Weber problem, where the customers are
continuously (uniformly) distributed along the boundary of a convex polygon. We
derive the closed-form expression for finding the average distance from a given
point to the continuously distributed customers along the boundary. A
Weiszfeld-type procedure is proposed for this model, which is shown to be
linearly convergent. We also derive a closed-form formula to find the average
distance for a given point to the entire convex polygon, assuming a uniform
distribution. Since the function is smooth, convex, and explicitly given, the
continuous version of the Fermat-Weber problem over a convex polygon can be
solved easily by numerical algorithms.
