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A Helly-type transversal theorem for n-dimensional unit balls
by
Gergely Ambrus
Auburn University
Coauthors: A. Bezdek, F. Fodor
Consider a family G of subsets of the Euclidean space Rn. A line l is called a transversal to the family if it intersects every element of G. Numerous results were proved for line transversals to families of convex sets in the plane, but in higher dimensions just a few facts are known. We prove the following theorem: Let F be a family of unit balls in Rn with the property that the mutual distances of the centers are at least 2√{2+ √2} ~ 3.6955 ... . If any 2n-1 members of F have a common line transversal, then F has a line transversal too. We also show that under the above distance condition our result is sharp.
Date received: October 21, 2005
Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # carr-08.