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

We show that for any compact convex body Qin the plane, the average distance from the Fermat-Weber center of Qto the points in Q is at most 99-50 root 3/36 . Delta(Q) < 0.3444 . Delta(Q), where Delta(Q) denotes the diameter of Q. This improves upon the previous bound of 2(4-root 3)/13 . Delta(Q) < 0.3490 Delta(Q). The average distance from the Fermat-Weber center of Qis calculated by comparing it with that of a circular sector of radius Delta(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 Qto the considered Fermat-Weber center are larger. A method for evaluating the average of all varied distances is given. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

It is shown 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 does not exceed a certain value, improving upon a previous bound.