Guarding galleries where every point sees a large area

We prove a conjecture of Kavraki, Latombe, Motwani and Raghavan that if X is a compact simply connected set in the plane of Lebesgue measure 1, such that any point x ∈ X sees a part of X of measure at least ε, then one can choose a set G of at most const 1/ε log 1/ε points in X such that any point of X is seen by some point of G. More generally, if for any k points in X there is a point seeing at least 3 of them, then all points of X can be seen from at most O(k³ log k) points.