Let S be a family of compact convex sets in R d . Let D(S) be the largest diameter of any member of S. The family 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) away from all d of the sets. We prove that if S is an "-separated family 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 rst Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one. 1 Introduction A k-transversal to a family S of point sets in R d is a k-at, i.e., an ane subspace of dimension k such as a point, line, or hyperplane, that intersects every member of S. We are interested in conditions under which a family of compact convex sets has a k-tran...