Illuminating high-dimensional convex sets (Q1900056)

Let \(S_1, \dots, S_n\) be mutually disjoint compact convex sets in Euclidean \(d\)-space \(E^d\). We say that the set \(S = S_1 \cup \cdots \cup S_n\) can be illuminated from a set \(L\) if for every boundary point \(x\) of \(S\) there is a point \(v \in L\) such that the segment joining \(v\) and \(x\) intersects \(S\) only at \(x\). The authors prove that for every compact convex set \(T\) in \(E^4\) there is a constant \(k\) such that the union of arbitrary \(n\) pairwise disjoint congruent copies of \(T\) can be illuminated from a set of \(kn\) points.