On the continuous Fermat-Weber problem for a convex polygon using Euclidean distance
Computational Geometry
2014-03-18 v1 Optimization and Control
Abstract
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.
Keywords
Cite
@article{arxiv.1403.3715,
title = {On the continuous Fermat-Weber problem for a convex polygon using Euclidean distance},
author = {Thomas T. C. K. Zhang and John Gunnar Carlsson},
journal= {arXiv preprint arXiv:1403.3715},
year = {2014}
}