Name
Abstract | ||
|---|---|---|
Let S be a finite collection of compact convex sets in \R d . Let D(S) be the largest diameter of any member of S . We say that the collection S is -separated if, for every 0 < k < d , any k of the sets can be separated from any other d-k of the sets by a hyperplane more than D(S)/2 away from all d of the sets. We prove that if S is an -separated collection of at least N( ) compact convex sets in \R d and every 2d+2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S . The number N( ) depends both on the dimension d and on the separation parameter . This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one. |
Year | DOI | Venue |
|---|---|---|
2001 | 10.1007/s00454-001-0016-0 | Discrete & Computational Geometry |
Keywords | DocType | Volume |
hyperplane transversals,well-separated convex set,helly-type theorem,convex set | Journal | 25 |
Issue | ISSN | ISBN |
4 | 0179-5376 | 1-58113-224-7 |
Citations | PageRank | References |
1 | 0.48 | 3 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Boris Aronov | 1 | 1430 | 149.20 |
| Jacob E. Goodman | 2 | 277 | 136.42 |
| Richard Pollack | 3 | 208 | 23.96 |
| Rephael Wenger | 4 | 441 | 43.54 |