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Discrete & Computational Geometry

, Volume 25, Issue 4, pp 507–517 | Cite as

A Helly-Type Theorem for Hyperplane Transversals to Well-Separated Convex Sets

  • B. Aronov
  • J. E. Goodman
  • R. Pollack
  • R. Wenger
Article

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 ɛ-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 ɛ D(S)/2 away from all d of the sets. We prove that if S is an ɛ -separated collection of at least N(ɛ) 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(ɛ) depends both on the dimension d and on the separation parameter ɛ . This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

Copyright information

© Springer-Verlag New York Inc. 2001

Authors and Affiliations

  • B. Aronov
    • 1
  • J. E. Goodman
    • 2
  • R. Pollack
    • 3
  • R. Wenger
    • 4
  1. 1.Polytechnic University, Brooklyn, NY 11201, USA aronov@ziggy.poly.eduUSA
  2. 2.City College, City University of New York, New York, NY 10031, USA jegcc@cunyvm.cuny.eduUSA
  3. 3.Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA pollack@cims.nyu.eduUSA
  4. 4.The Ohio State University, Columbus, OH 43210, USA wenger@cis.ohio-state.eduUSA

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