On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman

@article{Akopyan2018OnTC,
  title={On the Circle Covering Theorem by A.W. Goodman and R.E. Goodman},
  author={Arseniy V. Akopyan and Alexey Balitskiy and M. M. Grigorev},
  journal={Discrete \& Computational Geometry},
  year={2018},
  volume={59},
  pages={1001-1009}
}
In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii $$r_1$$r1, $$\ldots $$…, $$r_n$$rn in the plane, it is always possible to cover them by a disk of radius $$R = \sum r_i$$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… 
1 Citations
A Cap Covering Theorem
TLDR
If there is no great circle non-intersecting caps of $\mathcal S$ and dividing the set of caps into two non-empty subsets, then there is a cap of radius equal to the total radius of caps of $S$ covering all caps of S provided that the totalradius is less $\pi/2$.

References

SHOWING 1-7 OF 7 REFERENCES
On Non-separable Families of Positive Homothetic Convex Bodies
TLDR
It is proved that if B is a non-separable family of balls of radii with respect to an arbitrary norm in Rd, then B can be covered by a ball of radius 1nri and the conjecture that their theorem extends to arbitrary non- separable finite families of positive homothetic convex bodies is proved.
A Circle Covering Theorem
THEOREM 1. Let the circles C1, , C,, with radii ri, * , r,, lie in a plane and have the following property: No line of the plane divides the circles into two non-empty sets without touching or
Packing Convex Bodies by Cylinders
TLDR
Bounds are extended to the case of r-fold covering and packing and a packing analog of Falconer’s results are shown.
Nonseparable Convex Systems
Problems from St. Petersburg School Olympiad on Mathematics 2000-2002
  • Nevskiy Dialekt, St. Petersburg
  • 2006
Problems from St
  • Petersburg School Olympiad on Mathematics 2000–2002. Nevskiy Dialekt, St. Petersburg
  • 2006