On the circle covering theorem by A.W. Goodman and R.E. Goodman Journal Article


Author(s): Akopyan, Arseniy V; Balitskiy, Alexey; Grigorev, Mikhail
Article Title: On the circle covering theorem by A.W. Goodman and R.E. Goodman
Affiliation IST Austria
Abstract: In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erd\xC5\x91s: Given a family of (round) disks of radii (Formula presented.), (Formula presented.), (Formula presented.) in the plane, it is always possible to cover them by a disk of radius (Formula presented.), 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 (Formula presented.) with homothety coefficients (Formula presented.), it is always possible to cover them by a translate of (Formula presented.), provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.
Keywords: Goodman–Goodman theorem; Non-separable family; Positive homothets
Journal Title: Discrete & Computational Geometry
ISSN: 0179-5376
Publisher: Springer  
Date Published: 2017-03-02
Start Page: Epub ahead of Print
Copyright Statement: CC BY 4.0
DOI: 10.1007/s00454-017-9883-x
Notes: Open access funding provided by Institute of Science and Technology (IST Austria). The authors are grateful to Rom Pinchasi and Alexandr Polyanskii for fruitful discussions. Also the authors thank Roman Karasev, Kevin Kaczorowski, and the anonymous referees for careful reading and suggested revisions. The research of the first author is supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n◦[291734]. The research of the second author is supported by the Russian Foundation for Basic Research Grant 15-01-99563 A and Grant 15-31-20403 (mol_a_ved).
Open access: yes (OA journal)
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