On the continuous Fermat-Weber problem for a convex polygon using Euclidean distance

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.