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

Abstract

a b s t r a c tThe 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 in Q is larger than 1This proves a conjecture of Carmi, HarPeled and Katz. From the other direction, we prove that the same average distance is at most 2(4− √ 3) 13 · ∆(Q ) < 0.3490 · ∆(Q ). The new bound substantially improves the previous bound of 2 3 √ 3 · ∆(Q ) ≈ 0.3849 · ∆(Q ) due to Abu-Affash and Katz, and brings us closer to the conjectured value of 1 3 ·∆(Q ). We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.

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

“…The lower bound on c * was later improved from 1 7 to 4 25 [1]. Dumitrescu et al have eventually proved that c * = 1 6 [8], which confirms the conjecture due to Carmi, Har-Peled and Katz. Their work is based on two Steiner symmetrizations and a proof that the inequality c * ≥ 1 6 holds for a convex body with two orthogonal symmetry axes.…”

“…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].…”

“…Their work is based on two Steiner symmetrizations and a proof that the inequality c * ≥ 1 6 holds for a convex body with two orthogonal symmetry axes. The Steiner symmetrizations adopt in [8] preserve the area and the diameter, and do not increase the average distance from the corresponding Fermat-Weber centers.…”

“…a) E-mail: tan@wing.ncc.u-tokai.ac.jp of c * = 1 6 . As compared with the Steiner symmetrization method adopt in [8], the novelty of the method used in [1], [7] is its simplicity, because only elementary calculations (e.g., region partitions and recursive calculations) are needed. In this paper, we show that the simple method given in [1], [7] can be used to prove c * = 1 6 , too.…”

“…For instance, it is the ideal location for a fire or railroad station that serves the region Q. Although finding the Fermat-Weber center of Q is difficult, a simple, lineartime approximation scheme for the case where Q is a convex polygon is known [1], [8]. The result of c * = 1 6 helps give a better approximation ratio (Section 3).…”