On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman
Author(s)Akopyan, Arseniy; Balitskiy, Alexey; Grigorev, Mikhail
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Discrete & Computational Geometry
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.
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