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

Abstract

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 Tòth. 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 $\varDelta(Q)/3$.

This work was partially supported by the Grant-in-aid (MEXT/JSPS KAKENHI 23500024) for Scientific Research from Japan Society for the Promotion of Science and by National Natural Science Foundation of China under grant 61173034.