Piercing Balls Sitting on a Table by a Vertical Line

Let Fnbe a family of disjoint n balls all sitting on a fixed horizontal table T. Let ℓ denote a vertical line that meets T. We prove that if ℓ meets 2 k + 1 balls in Fn, then the radius of the smallest ball among the 2k + 1 balls is at most (2 − 3)ktimes the radius of the biggest ball among the 2 k + 1 balls. Using this result we prove that for anyFn the average number of balls an ℓ meets is at most logn + o(1). A similar result for a two-dimensional version is also given together with a lower bound of the least upper bound.