Cutting a Bunch of Grapes by a Plane

Let Dndenote a family of disjoint n disks in the plane. The max–min ratio λ of Dnis the ratio (the maximum radius)/(the minimum radius) among the disks in Dn. We prove that (1) If logλ = o(n), then there is a line both sides of which contain n / 2 − o(n) intact disks (such a line is called an almost-halving line), (2) for any constant c > 0, there is a family of disjoint n disks with logλ = cn that has no almost-halving line. The max–min ratio λ of a family Bnof disjoint n balls in R3is defined similarly. We also prove that (3) for any n ≥ 3, there is a family of disjoint n balls in R3such that every plane H in R3has a side that contains at most 2 intact balls of the family, and (4) if logλ = o((n / logn)1 / 3) for a family Bnof n disjoint balls in R3, then there is a plane both sides of which contain n / 2 − o(n) intact balls of Bn.