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Piercing a Set of Disjoint Balls by a Line

✍ Scribed by Hiroshi Maehara; Ai Oshiro

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
94 KB
Volume
94
Category
Article
ISSN
0097-3165

No coin nor oath required. For personal study only.

✦ Synopsis

Let F n denote a family of disjoint n balls in R d (d 2), and let *=*(F n ) denote the ratio (maximum radius)Γ‚(minimum radius) among the balls in F n . We prove that (1) there is a unit vector uÁ such that every line parallel to uÁ intersects at most O(-(1+log *) n log n) balls of F n , and (2) there is a family F n such that for any unit vector uÁ there is a line parallel to uÁ that intersects at least n&d+1 balls of F n .

πŸ“œ SIMILAR VOLUMES

Piercing Balls Sitting on a Table by a V
Piercing Balls Sitting on a Table by a Vertical Line
✍ Hiroshi Maehara; Ai Oshiro πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 127 KB

Let F n be a family of disjoint n balls all sitting on a fixed horizontal table T . Let denote a vertical line that meets T . We prove that if meets 2k + 1 balls in F n , then the radius of the smallest ball among the 2k + 1 balls is at most (2 -√ 3) k times the radius of the biggest ball among the