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

Simple Proof of the Lower Bound on the Average Distance from the Fermat-Weber Center of a Convex Body

“…Abu-Affash and Katz [1] showed that point o approximates the Fermat-Weber center of P , with µ P (o) µ * P . Combining Theorem 1 with Tan and Jiang's result on µ P (o) [10] gives us the approximation ratio (99 − 5 √ 3)/6 (< 2.07). This ratio may be further improved to 2 if µ P (o) ≤ ∆(P )/3 could be proved.…”

“…Abu-Affash and Katz were the first to show that c * 2 is between 2 3 √ 3 and 1 3 [1]. The upper bound on c * 2 was later improved to 2(4 − √ 3)/13 in [8], and further to (99 − 5 √ 3)/36 (< 0.3444) in [10]. Since the average distance between the points in a disk D and the Fermat-Weber center (i.e., the center) of D is ∆(D)/3, one may conjecture that c * 2 = 1 3 [7].…”