On the Helly Number for Hyperplane Transversals to Unit Balls |
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| Authors: | B. Aronov J. E. Goodman R. Pollack R. Wenger |
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| Affiliation: | (1) Polytechnic University, Brooklyn, NY 11201, USA aronov@ziggy.poly.edu , US;(2) City College, City University of New York, New York, NY 10031, USA jegcc@cunyvm.cuny.edu , US;(3) Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA pollack@geometry.cims.nyu.edu , US;(4) Ohio State University, Columbus, OH 43210, USA wenger@cis.ohio-state.edu, US |
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| Abstract: | We prove two results about the Hadwiger problem of finding the Helly number for line transversals of disjoint unit disks in the plane, and about its higher-dimensional generalization to hyperplane transversals of unit balls in d -dimensional Euclidean space. These consist of (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks—thus correcting a 40-year-old error; and (b) a lower bound of d+3 on the Helly number for hyperplane transversals to suitably separated families of unit balls in R d . Received January 25, 1999, and in revised form July 7, 1999. |
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