Improved bounds on the average distance to the Fermat–Weber center of a convex object

Abstract

We show that for any convex object Q in the plane, the average distance between the Fermat-Weber center of Q and the points in Q is at least 4∆(Q)/25, and at most 2∆(Q)/(3 √ 3), where ∆(Q) is the diameter of Q.We use the former bound to improve the approximation ratio of a load-balancing algorithm of Aronov et al. [1].

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

“…Since ∆(P ) = 2, we have µ * P > 1 6 · ∆(P ), as desired. ✷ Lemma 4 Let T be a triangle in the first quadrant with a vertical side on the line x = a, where 0 ≤ a < 1, and a third vertex at (1,0).…”

“…Since the average distance from the Fermat-Weber center of Q is not larger than that from o, we immediately get the same upper bound on c 2 . We need the next simple lemma established in [1]. Its proof follows from the definition of average distance.…”

“…Lemma 5 [1]. Let Q 1 , Q 2 be two (not necessarily convex) disjoint bodies in the plane, and p be a point in the plane.…”

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

“…It is well known that FW Q ∈ Q, if Q is convex [1]. Denote by c * the infimum of µ * Q /∆(Q) over all convex bodies, where ∆(Q) denotes the diameter of Q. Carmi, Har-Peled and Katz were the first to show that 1 7 ≤ c * ≤ 1 6 , and they conjectured that c * = 1 6 , because a flat rhombus Q can be constructed such that µ * Q tends to ∆(Q ) 6 [7]. The lower bound on c * was later improved from 1 7 to 4 25 [1].…”

Serving regional demand in facility location

“…() proved that for a convex demand region the average distance from the optimal facility location is within [ d /7, d /7], where d is the region diameter. Abu‐Affash and Katz () improved the bounds to [4 d /25, ]. More recently, Puerto and Rodríguez‐Chía () obtained the geometrical characterizations of the entire set of optimal solutions given demand defined by some probability distribution.…”