On the Fermat--Weber center of a convex object

Authors:
Paz Carmi;Sariel Har-Peled;Matthew J. Katz
Affiliations:
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel;Department of Computer Science, University of Illinois, 201 N. Goodwin Avenue, Urbana, IL 61801, USA;Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Venue:
Computational Geometry: Theory and Applications
Year:
2005

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Abstract

We show that for any convex object Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at least @D(Q)/7, where @D(Q) is the diameter of Q, and that there exists a convex object P for which this distance is @D(P)/6. We use this result to obtain a linear-time approximation scheme for finding an approximate Fermat-Weber center of a convex polygon Q.