On the upper bound on the average distance from the Fermat-Weber center of a convex body

Abstract We show that for any compact convex body Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at most 99 − 50 3 36 ⋅ Δ ( Q ) 0.3444 ⋅ Δ ( Q ) , where Δ ( Q ) denotes the diameter of Q. This improves upon the previous bound of 2 ( 4 − 3 ) 13 ⋅ Δ ( Q ) 0.3490 ⋅ Δ ( Q ) . The average distance from the Fermat-Weber center of Q is calculated by comparing it with that of a circular sector of radius Δ ( Q ) / 2 , whose area is the same as that of Q. As compared to the points of that circular sector, the distances of some points of Q to the considered Fermat-Weber center are larger. A method for evaluating the average of all varied distances is given.