On the circle covering theorem by A.W. Goodman and R.E. Goodman
Akopyan, Arseniy
Balitskiy, Alexey
Grigorev, Mikhail
ddc:516
ddc:000
In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, … , rn in the plane, it is always possible to cover them by a disk of radius R= ∑ ri, 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⊂ Rd with homothety coefficients τ1, … , τn> 0 , it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.
Springer
2018
info:eu-repo/semantics/article
doc-type:article
text
http://purl.org/coar/resource_type/c_6501
https://research-explorer.ista.ac.at/record/1064
https://research-explorer.ista.ac.at/download/1064/5844
Akopyan A, Balitskiy A, Grigorev M. On the circle covering theorem by A.W. Goodman and R.E. Goodman. <i>Discrete & Computational Geometry</i>. 2018;59(4):1001-1009. doi:<a href="https://doi.org/10.1007/s00454-017-9883-x">10.1007/s00454-017-9883-x</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00454-017-9883-x
info:eu-repo/semantics/altIdentifier/issn/01795376
info:eu-repo/semantics/altIdentifier/issn/14320444
info:eu-repo/grantAgreement/EC/FP7/291734
info:eu-repo/semantics/openAccess