Separation of Convex Sets

A line L separates a set A from a collection S of plane sets if A is contained in one of the closed half-planes defined by L, while every set in S is contained in the complementary closed half-plane. Let f(n) be the largest integer such that for any collection F of n closed disks in the plane with pairwise disjoint interiors, there is a line that separates a disk in F from a subcollection of F with at least f(n) disks. In this note we prove that there is a constant c such that f(n) > (n−c) 2 . An analogous result for the d-dimensional Euclidean space is also discussed. A line L separates a set A from a collection S of plane sets if A is contained in one of the closed half-planes defined by L, while every set in S is contained in the complementary closed half-plane. Alon et al. proved in [1] that there is a constant a > 0 such that, for any collection F of n congruent disks in the plane with pairwise disjoint interiors, there is a line L that leaves at least n 2 − a √ (n lnn) disks of F on each closed half-plane defined by L. We denote by f(n) the largest integer such that for any collection F of n closed disks in the plane with pairwise disjoint interiors and arbitrary radii, there is a line that separates a disk in F from a subcollection of F with at least f(n) disks. Czyzowicz et al. proved in [2] that n 2 ≥ f(n) ≥ (n−7) 4 . In this note we prove that there is a constant c such that f(n) ≥ (n−c) 2 . Let A be a compact convex set in the plane with nonempty interior, we denote by e(A) the ratio D(A) r(A) , where D(A) is the diameter of A and r(A) is the radius of the largest disk inscribed in A. We prove the following stronger result: Département d’lnformatique, Université du Québec à Hull, Hull, Qué., Canada Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, México D.F. México Computer Science, University of Ottawa, Ottawa, ON. Canada