On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman
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454_2017_Article_9883.pdf
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Author(s)
Akopyan, Arseniy
Balitskiy, Alexey
Grigorev, Mikhail
Date Issued
March 2017
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Akopyan, Arseniy, Alexey Balitskiy, and Mikhail Grigorev. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” Discrete & Computational Geometry (2017): n. pag.
Version
Final published version
Abstract
In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r[subscript 1],..., r[subscript n] in the plane, it is always possible to cover them by a disk of radius R = ∑r[subscript i], provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K ⊂ R[superscript d] with homothety coefficients τ[subscript 1],..., τ[subscript n] > 0, it is always possible to cover them by a translate of d+1/2 (∑τ[subscript 1]) K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00454-017-9883-x
