Helly-type theorem for hyperplane transversals to well-separated convex sets

Boris Aronov, Jacob E. Goodman, Richard Pollack, Rephael Wenger

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Let S be a family of compact convex sets in Rd. Let D(S) be the largest diameter of any member of S. The family S is ε-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 ε/D(S) away from all d of the sets. We prove that if S is an ε-separated family of at least N(ε) compact convex sets in Rd 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(ε) depends both on the dimension d and on the separation parameter ε. This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

Original language English (US) Proceedings of the Annual Symposium on Computational Geometry ACM 57-63 7 Published - 2000 16th Annual Symposium on Computational Geometry - Hong Kong, Hong KongDuration: Jun 12 2000 → Jun 14 2000

Other

Other 16th Annual Symposium on Computational Geometry Hong Kong, Hong Kong 6/12/00 → 6/14/00

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Helly-type Theorems
Transversals
Hyperplane
Convex Sets
Compact Convex Set
Arbitrary
Family

ASJC Scopus subject areas

• Chemical Health and Safety
• Software
• Safety, Risk, Reliability and Quality
• Geometry and Topology

Cite this

Aronov, B., Goodman, J. E., Pollack, R., & Wenger, R. (2000). Helly-type theorem for hyperplane transversals to well-separated convex sets. In Proceedings of the Annual Symposium on Computational Geometry (pp. 57-63). ACM.

Helly-type theorem for hyperplane transversals to well-separated convex sets. / Aronov, Boris; Goodman, Jacob E.; Pollack, Richard; Wenger, Rephael.

Proceedings of the Annual Symposium on Computational Geometry. ACM, 2000. p. 57-63.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Aronov, B, Goodman, JE, Pollack, R & Wenger, R 2000, Helly-type theorem for hyperplane transversals to well-separated convex sets. in Proceedings of the Annual Symposium on Computational Geometry. ACM, pp. 57-63, 16th Annual Symposium on Computational Geometry, Hong Kong, Hong Kong, 6/12/00.
Aronov B, Goodman JE, Pollack R, Wenger R. Helly-type theorem for hyperplane transversals to well-separated convex sets. In Proceedings of the Annual Symposium on Computational Geometry. ACM. 2000. p. 57-63
Aronov, Boris ; Goodman, Jacob E. ; Pollack, Richard ; Wenger, Rephael. / Helly-type theorem for hyperplane transversals to well-separated convex sets. Proceedings of the Annual Symposium on Computational Geometry. ACM, 2000. pp. 57-63
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