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

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. Received August 10, 2000, and in revised form January 24, 2001. Online publication April 6, 2001.