10.1007/s00454-017-9883-x
Akopyan, Arseniy
Arseniy
Akopyan
Balitskiy, Alexey
Alexey
Balitskiy
Grigorev, Mikhail
Mikhail
Grigorev
On the circle covering theorem by A.W. Goodman and R.E. Goodman
Springer
2018
2018-12-11T11:49:57Z
2020-01-21T08:46:05Z
journal_article
https://research-explorer.app.ist.ac.at/record/1064
https://research-explorer.app.ist.ac.at/record/1064.json
01795376
482518 bytes
application/pdf
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.