Sciweavers

Share
ISAAC
2009
Springer

New Bounds on the Average Distance from the Fermat-Weber Center of a Planar Convex Body

10 years 4 days ago
New Bounds on the Average Distance from the Fermat-Weber Center of a Planar Convex Body
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 · ∆(Q), where ∆(Q) is the diameter of Q. This 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.
Adrian Dumitrescu, Csaba D. Tóth
Added 26 May 2010
Updated 26 May 2010
Type Conference
Year 2009
Where ISAAC
Authors Adrian Dumitrescu, Csaba D. Tóth
Comments (0)
books