A New Approach to the Upper Bound on the Average Distance from the Fermat-Weber Center of a Convex Body

We show that for any convex body Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at most \(\frac{99-50 \sqrt{3}}{36} \cdot \varDelta(Q) < 0.3444 \cdot \varDelta(Q)\), where \(\varDelta(Q)\) denotes the diameter of Q. This improves upon the previous bound of \(\frac{2 (4 - \sqrt{3})}{13} \cdot \Delta(Q) \approx 0.3490 \cdot \varDelta(Q)\), due to Dumitrescu, Jiang and Toth. Our new method to evaluate the average distance from the Fermat-Weber center of Q is to transform Q into a circular sector of radius \(\varDelta(Q)/2\). Some points of Q may decrease their distances to the Fermat-Weber center in Q after the transformation, but the total amount of varied distances can be well controlled. Our work sheds more light on the conjectured upper bound \(\varDelta(Q)/3\).