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 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 εD(S)/2 away from all d of the sets. We prove that if S is an ε-separated collection of at least N (ε) 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 (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.