Common transversals in the plane: The fractional perspective

Abstract

A fresh look is taken at the fractional Helly theorem for line transversals to families of convex sets in the plane. This theorem was first proved in 1980 by Katchalski and Liu [M. Katchalski, A. Liu, Symmetric twins and common transversals, Pacific J. Math. 86 (1980) 513–515]. It asserts that for every integer k ≥ 3 , there exists a real number ρ ( k ) ∈ ( 0 , 1 ) such that the following holds: If K is a family of n compact convex sets in the plane, and any k or fewer members of K have a line transversal, then some subfamily of K of size at least ρ ( k ) n has a line transversal. A lower bound on ρ ( k ) is obtained which is stronger than the one obtained in [M. Katchalski, A. Liu, Symmetric twins and common transversals, Pacific J. Math. 86 (1980) 513–515]. Also, examples are given to show that a conjecture of Katchalski concerning the value of ρ ( 3 ) , if true, is the best possible.