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

article On the circle covering theorem by A.W. Goodman and R.E. Goodman published yes Arseniy Akopyan author 430D2C90-F248-11E8-B48F-1D18A9856A870000-0002-2548-617X Alexey Balitskiy author Mikhail Grigorev author HeEd department International IST Postdoc Fellowship Programme project 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. https://research-explorer.ista.ac.at/download/1064/5844/2018_DiscreteComp_Akopyan.pdf application/pdfno Springer2018 eng Discrete & Computational Geometry 0179-5376 1432-0444 00043220550001110.1007/s00454-017-9883-x 5941001-1009 Akopyan A, Balitskiy A, Grigorev M. 2018. On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete & Computational Geometry. 59(4), 1001–1009. Akopyan, A., Balitskiy, A., & Grigorev, M. (2018). On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete & Computational Geometry. Springer. <a href="https://doi.org/10.1007/s00454-017-9883-x">https://doi.org/10.1007/s00454-017-9883-x</a> Akopyan, Arseniy, et al. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” Discrete & Computational Geometry, vol. 59, no. 4, Springer, 2018, pp. 1001–09, doi:<a href="https://doi.org/10.1007/s00454-017-9883-x">10.1007/s00454-017-9883-x</a>. Akopyan A, Balitskiy A, Grigorev M. On the circle covering theorem by A.W. Goodman and R.E. Goodman. Discrete & Computational Geometry. 2018;59(4):1001-1009. doi:<a href="https://doi.org/10.1007/s00454-017-9883-x">10.1007/s00454-017-9883-x</a> A. Akopyan, A. Balitskiy, M. Grigorev, Discrete & Computational Geometry 59 (2018) 1001–1009. Akopyan, Arseniy, Alexey Balitskiy, and Mikhail Grigorev. “On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman.” Discrete & Computational Geometry. Springer, 2018. <a href="https://doi.org/10.1007/s00454-017-9883-x">https://doi.org/10.1007/s00454-017-9883-x</a>. A. Akopyan, A. Balitskiy, and M. Grigorev, “On the circle covering theorem by A.W. Goodman and R.E. Goodman,” Discrete & Computational Geometry, vol. 59, no. 4. Springer, pp. 1001–1009, 2018. 10642018-12-11T11:49:57Z2026-05-20T10:19:33Z