On the helly number for hyperplane transversals to unit balls

B. Aronov; J. E. Goodman; R. Pollack; R. Wenger

doi:10.1007/s004540010024

On the helly number for hyperplane transversals to unit balls

B. Aronov, J. E. Goodman, R. Pollack, R. Wenger

Research output: Contribution to journal › Article › peer-review

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.

Original language	English (US)
Pages (from-to)	171-176
Number of pages	6
Journal	Discrete and Computational Geometry
Volume	24
Issue number	2-3
DOIs	https://doi.org/10.1007/s004540010024
State	Published - 2000

ASJC Scopus subject areas

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1007/s004540010024

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AU - Aronov, B.

AU - Goodman, J. E.

AU - Pollack, R.

AU - Wenger, R.

PY - 2000

Y1 - 2000

N2 - 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 ℝd.

AB - 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 ℝd.

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ER -

On the helly number for hyperplane transversals to unit balls

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