On optimal disc covers and a new characterization of the Steiner center (Q283866)

The authors consider the problem of sphere coverage in a \(d\)-dimensional Euclidean space. In the considered problem they have a set of \(N\) points \(P=\{P_1,P_2,\dots,P_N\}\) in \(\mathbb R^d\) and they take an arbitrary point \(\Omega\in \mathbb R^d\). They define spheres \(S_{P_i}(\Omega)\) as the spheres having the line segments \([\Omega P_i]\) as diameters for \(i=1,2,\dots,N\) and consider the union of the spheres. The authors first prove that the resulting shape of this union covers the convex hull \(CH(P)\) for all \(\Omega\in \mathbb R^d\). Next, they ask a question: What is the location \(\Omega^\ast\) which minimizes the excess volume and hence the total volume of the spheres' union? The authors give a proof that \(\Omega^\ast\) is the so-called Steiner center of \(CH(P)\), i.e., a weighted centroid of the vertices of a convex polygon, where the weights are proportional to the exterior angles at the vertices. The proof is made only for the case \(d=2\). For \(d>2\), the authors conjecture that a similar results holds.