Cutting a Bunch of Grapes by a Plane

✍ Scribed by Hiroshi Maehara; Ai Oshiro

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 110 KB
• Volume: 22
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.2001.0452

✦ Synopsis

Let D n denote a family of disjoint n disks in the plane. The max-min ratio λ of D n is the ratio (the maximum radius)/(the minimum radius) among the disks in D n . We prove that (1) If log λ = o(n), then there is a line both sides of which contain n/2 -o(n) intact disks (such a line is called an almost-halving line), (2) for any constant c > 0, there is a family of disjoint n disks with log λ = cn that has no almost-halving line. The max-min ratio λ of a family B n of disjoint n balls in R 3 is defined similarly. We also prove that (3) for any n ≥ 3, there is a family of disjoint n balls in R 3 such that every plane H in R 3 has a side that contains at most 2 intact balls of the family, and (4) if log λ = o((n/ log n) 1/3 ) for a family B n of n disjoint balls in R 3 , then there is a plane both sides of which contain n/2 -o(n) intact balls of B n .