New bounds on the average distance from the Fermat-Weber center of a planar convex body

TL;DR

This paper establishes new bounds on the average distance from the Fermat-Weber center to points in a convex body, improving previous bounds and confirming conjectures for symmetric cases.

Contribution

It proves tighter bounds on the average distance from the Fermat-Weber center to convex bodies, confirming a conjecture for symmetric bodies and advancing understanding of geometric center properties.

Findings

Lower bound: average distance > (1/6) * diameter

Upper bound: average distance < 0.3490 * diameter

Confirmed conjecture for centrally symmetric convex bodies

Abstract

The Fermat-Weber center of a planar body $Q$ is a point in the plane from which the average distance to the points in $Q$ is minimal. We first show that for any convex body $Q$ in the plane, the average distance from the Fermat-Weber center of $Q$ to the points of $Q$ is larger than ${1/6} \cdot \Delta(Q)$, where $\Delta(Q)$ is the diameter of $Q$. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most $\frac{2(4-\sqrt3)}{13} \cdot \Delta(Q) < 0.3490 \cdot \Delta(Q)$. The new bound substantially improves the previous bound of $\frac{2}{3 \sqrt3} \cdot \Delta(Q) \approx 0.3849 \cdot \Delta(Q)$ due to Abu-Affash and Katz, and brings us closer to the conjectured value of ${1/3} \cdot \Delta(Q)$. We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.